CONTROL AND DYNAMIC SYSTEMS
Advances in Theory and Applications Volume 74
CONTROL AND DYNAMIC SYSTEMS
Advances in Theory and Applications Volume 74
CONTRIBUTORS TO THIS VOLUME DE ABRE U-GAR CIA J. CHEN M UNTHER A. DAHLEH SARIT K. DAS ANTHONY N. MICHEL AHMAD A. MOHAMMED YAS UHIKO MUTOH PETER N. NIKIFORUK R. J. PATTON MILOJE S. RADENKOVIC P. K. RAJAGOPALAN TIELONG SHEN ROBERT E. SKELTON PETROS G. VOULGARIS G UOMING G. ZHU
J. A .
CONTROL AND DYNAMIC SYSq'EMS ADVANCES IN THEORY AND APPLICATIONS
Edited by
CORNELIUS T. LEONDES School of Engineering and Applied Science University of California, Los Angeles Los Angeles, California
VOLUME 74:
TECHNIQUES IN DISCRETE AND CONTINUOUS ROBUST SYSTEMS
ACADEMIC PRESS San Diego New York Boston London Sydney Tokyo Toronto
This book is printed on acid-free paper. ( ~ Copyright 9 1996 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.
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PRINTED IN THE UNITED STATES OF AMERICA 96 97 98 99 00 01 QW 9 8 7 6 5
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1
CONTENTS
CONTRIBUTORS .................................................................................. PREFACE ................................................................................................
vii ix
Optimal and Robust Controllers for Periodic and Multirate Systems ......................................................................................
Petros G. Voulgaris and Munther A. Dahleh ................................
59
A Two-Riccati, Feasible Algorithm for Guaranteeing Output L~ Constraints ............................................................................
97
Discrete-Time Robust Adaptive Control Systems
Miloje S. Radenkovic and Anthony N. Michel
Guoming G. Zhu and Robert E. Skelton Techniques of Analysis and Robust Control via Zero-Placement of Periodically Compensated Discrete-Time Plants ................................... 133
Sarit K. Das and P. K. Rajagopalan Robust Fault Detection and Isolation (FDI) Systems
........................... 171
R. J. Patton and J. Chen Absolute Stability of Discrete Nonlinear Feedback Systems
............... 225
Yashuhiko Mutoh, Tielong Shen, and Peter N. Nikiforuk
vi
CONTENTS
Continuous Time and Discrete Time L y a p u n o v Equations" R e v i e w and N e w Directions ................................................................................ 253
Ahmad A. Mohammed and J. A. De Abreu-Garcia
I N D E X ..................................................................................................... 305
CONTRIBUTORS
Numbers in parentheses indicatethe pages on which the authors' contributions begin. J. A. De Abreu-Garcia (253), Department of Electrical Engineering, The
University of Akron, Akron, Ohio 44325 J. Chen (171), Division of Dynamics and Control, University of Strathclyde,
Glasgow, Gl lXJ, United Kingdom Munther A. Dahleh (1), Laboratory of Information and Decision Systems,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Sarit K. Das (133), Department of Electrical Engineering, Indian Institute
of Technology, Kharagpur 721302, India Anthony N. Michel (59), Department of Electrical Engineering, University
of Notre Dame, Notre Dame, Indiana 46556 Ahmad A. M o h a m m e d (253), Department of Electrical Engineering, The
University of Akron, Akron, Ohio 44325 Yasuhiko Mutoh (225), Department of Mechanical Engineering, Sophia Uni-
versity, Tokyo, Chiyoda-ku, Japan Peter N. Nikiforuk (225), Department of Mechanical Engineering, Univer-
sity of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N OWO R. J. Patton (171), Department of Electronic Engineering, University of Hull,
Hull HU6 7RX, United Kingdom Miloje S. Radenkovic (59), Department of Electrical Engineering, Univer-
sity of Colorado at Denver, Denver, Colorado 80217 P. K. Rajagopalan (133), Department of Electrical Engineering, Indian In-
stitute of Technology, Kharagpur 721302, India vii
viii
CONTRIBUTORS
Tielong Shen (225), Department of Electrical Engineering, Sophia University, Tokyo, Chiyoda-ku, Japan Robert E. Skelton (97), Space Systems Control Laboratory, Purdue University, West Lafayette, Indiana 47907 Petros G. Voulgaris (1), Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 Guoming G. Zhu (97), Cummins Engine Company, Inc., Columbus, Indiana 47202
PREFACE Effective control concepts and applications date back over millennia. One very familiar example of this is the windmill. It was designed to derive maximum benefit from windflow, a simple but highly effective optimization technique. Harold Hazen's 1932 paper in the Journal of the Franklin Institute was one of the earlier reference points wherein an analytical framework for modem control theory was established. There were many other notable items along the way, including the MIT Radiation Laboratory series volume on servomechanisms, the Brown and Campbell book, Principles of Servomechanisms, and Bode's book, Network Analysis and Synthesis Techniques, all published shortly after mid-1945. However, it remained for Kalman's papers of the late 1950s (wherein a foundation for modem state-space techniques was established) and the tremendous evolution of digital computer technology (which was founded on the continuous giant advances in integrated electronics) to establish truly powerful control systems techniques for increasingly complex systems to be developed. Today we can look forward to a future that is rich in possibilities in many systems of major significance: manufacturing, electric power, robotics, aerospace, and others with significant economic, safety, cost, and reliability implications. Thus, this volume is devoted to the most timely theme of "Techniques in Discrete and Continuous Robust Systems." The first contribution to this volume is "Optimal and Robust Controllers for Periodic and Multirate Systems," by Petros G. Voulgaris and Munther A. Dahleh. The problems of optimal disturbance rejection and robust stability in periodic and multirate systems are treated in this contribution, and techniques for dealing with these in several major cases are presented. This contribution further notes that the problem of robust stabilization in periodic and multirate systems can be treated without introducing conservatism by considering the problem for the equivalent linear time invariant (LTI) system. Other important results and techniques are also discussed, and as such it is a most appropriate chapter to begin this volume. The second contribution is "Discrete-Time Robust Adaptive Control Systems," by Miloje S. Radenkovic and Anthony N. Michel. The robust adaptive control problem has been the topic of numerous studies, publications,
x
PREFACE
and discussions over the past decade, and it has been observed that algorithms designed for the case of perfect system models cannot provide global stability of the adaptive system in the presence of unmodeled dynamics and external disturbances. Techniques are presented for establishing the global stability of adaptive systems for unmodeled system dynamics and external disturbances which are unstructured, complex, and large at high frequencies. It is worth noting that results similar to those presented in this contribution can be established for the general delay case. The next contribution is "A Two-Riccati, Feasible Algorithm for Guaranteeing Output L~ Constraints," by Guoming G. Zhu and Robert E. Skelton. This contribution presents a rather powerful new and computationally efficient methodology for designing measurement or dynamic controllers that guarantee requisite output bounds for measures of bounded input disturbances while minimizing a weighted summation of measures of the upper bound of the outputs for each control channel. The substantial effectiveness of the results presented in this contribution is illustrated by several examples. Techniques for the design of a linear, discrete periodic controller structure that has the maximum possible degrees of freedom for its order are presented in "Techniques of Analysis and Robust Control via Zero-Placement of Periodically Compensated Discrete-Time Plants," by Sarit K. Das and R. K. Rajagopalan. Analytical methods for dealing with periodic systems and the zero placement capability of periodic controllers are also presented. The role of periodic controllers in gain margin improvement of unstable plants with nonminimum phase zeros via the zero placement approach and periodic controllers is treated. Numerous examples demonstrate the effectiveness of the techniques. "Robust Fault Detection and Isolation (FDI) Systems," by R. J. Patton and J. Chen, is an in-depth treatment of the techniques in the field of robustness for fault diagnosis that rest on the model-based residual generation methods (which are defined in this contribution). This approach to fault detection and isolation offers rather significant potential, particularly in view of the role it can and probably will play in future industrial systems, process plants, and other applications of major significance. The next contribution, "Absolute Stability of Discrete Nonlinear Feedback Systems," by Yasuhiko Mutoh, Tielong Shen, and Peter N. Nikiforuk, presents various approaches for the establishment of the stability of discretetime nonlinear feedback systems. Techniques are presented for both singleinput-single-output (SISO) systems and multiple-input-multiple-output (MIMO) systems. Examples illustrate the utility of the methods presented. The final contribution to this volume is "Continuous Time and Discrete Time Lyapunov Equations: Review and New Directions," by Ahmad A. Mohammad and J. A. De Abreu-Garcia. The fundamental reasons for robust design techniques include among other factors the maintenance of satisfactory system performance in the face of model uncertainties and, perhaps more
PREFACE
xi
importantly, the maintenance of system stability. This contribution is an indepth treatment of one of the most powerful means for confirming stability for both linear and nonlinear systems. As such this is a most suitable contribution with which to conclude this volume. The contributors to this volume are all to be highly commended for their contributions to this comprehensive treatment of techniques in discrete and continuous robust systems. They have produced a work which should provide a unique and useful reference on this broad subject internationally for years to come.
This Page Intentionally Left Blank
Optimal and Robust Controllers for Periodic and Multirate Systems Petros G. Voulgaris Coordinated Science Laboratory University of Illinois at Urb~na-Chmmpaign
Munther A. Dahleh L a b o r a t o r y of I n f o r m a t i o n ~nd D e c i s i o n S y s t e m s
Massachusetts Institute of Technology
I. I N T R O D U C T I O N The study of periodically time varying systems is a topic great practical and theoretical importance. In [13] an equivalence between m-input, p-output, linear, N-periodic, causal, discrete-time systems and a class of discrete-time linear, time invariant, causal systems was established. Namely, this class consists of ruN,input, pN-output, linear time invariant (LTI) systems with A- transforms P(A) such that P(0) is a block lower triangular matrix. This equivalence is strong in the sense that it preserves the algebraic structure (isomorphism) and the norm (isometry). This equivalence is termed "lifti n ~ and the LTI system that lifting associates with the N-periodic system is called the ~lifted" system. Hence, we can effectively use the theory of LTI systems to study periodic ones. In fact, the authors in [13] use this equivalence to prove that although the performance is not improved , periodic compensators for LTI plants'offer significant advantages in terms of robustness to parametric uncertainty. In this chapter, we define the problem of optimal disturbance rejection in periodic systems and present solutions to the following three cases: I. Optimal L~~ to t~176disturbance rejection. 2. Optimal t2 to t2 disturbance rejection. 3. Optimal rejection of stochastic disturbances (the L Q G problem). Utilizing the power of lifting and the results in [3] we can easily infere that in all three cases the optimal controller for the N-periodic system can be obtained by solving the equivalent LTI problem. This problem however, includes a constraint on the optimal LTI compensator C(A), namely CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
2
PETROS G. V O U L G A R I S A N D M U N T H E R A. D A H L E H
C(0) should be block lower triangular matrix so that C corresponds to a causal N-periodic controller. This constraint makes the optimization problem a non-standard one and its solution is the theme of sections IV and V. Furthermore in section VI we demonstrate that the problem of optimal disturbance rejection for multirate sampled systems can be treated analogously. In particular, we show how with a simple modification the same approach can be used to obtain the optimal multirate compensator. Also in this chapter, we consider the problem of robust stabilization in periodic and multirate plants. We indicate that this problem can be analyzed without introducing conservatism by considering the same problem for the equivalent LTI system.
II. M A T H E M A T I C A L
PRELIMINARIES
This section presents the notation and definitions to be used throughout the chapter. Also, some important to our development mathematical results are provided. References are given to cover all of the needed mathematical background.
1
Generic
Notation
In this section we give some generic notation that is used throughtout the thesis.
p(A)
The spectral radius of the matrix A.
~[A]
The maximum singular value of the matrix A.
Izlp
The p-norm of the finite dimensional vector z = (zl z~ . . . z , ) T given as
Ixl,, = ()":~ Ix, I) ~/', p < oo i=l
Izlp = m.,~ Ix, I, p = oo. $
Ial~
The 1-norm of the m x n matrix A = (Aij) given as
FJ
IAII= /t(A)
max
i--- 1,...,m
~'~IA,#Ij=0
The A-transform of a m x n real sequence H = {H(k)}~~ defined as: oo
9(~) = ~
k~---oo
#(k)~ ~
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
X
I
The dual space of the normed linear space X.
BX
The closed unit ball of X.
•
The left annihilator of S C X*.
S•
The right annihilator of S C X.
IIs
The projection operator onto the subset S of the Hilbert space X.
(Z~
T*
n~
Z*)
The value of the bounded linear functional z" at point z ~ X. The adjoint of the operator T. The kth-truncation operator acting on a m x 1
~e~tor~u~d ~equen~ {~(~)}~~ a~ Hkm 9{u(0), u(1),...}
Am
: {u(0),..., u(k), 0, 0,...}.
The right shift operator acting on a m x 1 vector valued sequence {u(k) } ~~ as
A m : {u(0), u(1),...}
2
3
Some
, {0, u(0), u(1),...}.
Basic Spaces
In this section we define certain important normed linear spaces that we very frequently refer to in the course of our development. These spaces are the following (for details look at [22,16,14,28]):
l~x" 9The Banach space of all m x n matrices H each of whose entries is a right sided, absolutely summable real sequence Hij = {Hij (k) }~~ The norm is defined as:
IIHII~,
mXs
n dell" =
max E i
oo
~ IHij(t)l.
j=l k=0
: The Banach space of real rn x 1 vectors u each of whose components is a magnitude bounded reM sequence { u , ( k ) } r : 0. The norm is defined as:
II~llt: d.j miax(s~pI~,(k)l).
4
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
~~ 9The extended t~~ space: it is the space of all real right sided m x 1 vector valued sequences.
t ~ 9 The Hilbert space of real rn • 1 vectors u each of whose components is an energy bounded real sequence {u~ (h) }~~ The norm is defined as: m
oo
i=1 k=O
x , " The real Banach space of all m x n matrices H(A) such t h a t H(A) is the A-transform of an I x , x , sequence H. The norm is defined as
This space is isometrically isomorphic to lx.,x, i.e., l ~ x . _-- . A . , x . .
.A~x n 9 The Banach space of all m x n matrices H each of whose entries is a right sided, magnitude bounded real sequence Hij = {Hij(k))~~ The n o r m is defined as: m
IIHII.A:.,,..
def
max(sup IH,i(k)l).
=
dfX
cox. zero.
9The subspace of ~ x .
J
k
consisting of all elements which converge to
O0
s " The Banach space of all m x . m a t r i x valued functions F defined on the unit circle of the complex plain with def
I]F][s = ess sup ~'[F(eJ')]< oo. ,e[0,~.]
oo
7~mx. 9 The Banach space of all rn x n matrix valued functions F analytic in the open unit disk of the complex plain with def
IIFII - =
sup
max ~ [ F ( r e J ' ) ] < oo.
rE[O,1) #El0,2-]
This space can be considered as a closed subspace of s
oo
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
5
s 2 1 5 " The Hilbert space of matrix valued functions F defined on the unit circle of the complex plain with
IIFII~ def = [(2x) -1 fo 2" t r a c e ( F T ( e - { ' ) F ( e ~ ' ) ) d e ]
~/~ < oo.
~/~xn " The Hilbert space of all m x n matrix valued functions F analytic in the open unit disk of the complex plane with def
sup [(2r -1 JO02r t r a c e ( F T ( r e - i ' ) F ( r e ~ ' ) ) d O ] ~/2 < oo. re[o,1)
This space can be considered as a closed subspace of s
t 2 = ~2,,1. 3
Input-Output Systems
Characterization
Moreover,
of Linear
In this section we consider the input-output characterization of systems by viewing them as linear operators. 3.1
Causality
We start with the notion of a causal operator D e f i n i t i o n 3.1 L e t T : l~ ,e
n Tu =
be an operator. T is called causal i f
, ~,e
ThOu,
vk = o,
2,...,
T is called s t r i c t l y causal i f II ~ T u = IIk - m -T- InI k - I u,
V k = O , 1,2, . . . .
The class of all causal operators T on/oo,e will be denoted by s215 Such operators can be represented by infite block lower triangular matrices (Toeplitz) of the form
To(O) o "
i
""/
6
PETROS G. V O U L G A R I S AND M U N T H E R A. D A H L E H
where Tk(i) axe m • n matrices for all i, k. This representation is another way to state that these operators are convolution operators; i.e., if y = Tu then k
~(k)
= ~
n(k
-
i)~(~)
i----0
3.2 3.2.1
Stability T h e spaces ~ " ( P ) ,
~"
(t 2)
Next, we consider the notion of/a-stability where ~ = or, 2. D e f i n i t i o n 3.2 Let T be a causa/operator in s if its induced norm over t a is bounded; i.e., i f IITII
=
sup ~,:.~,o Ilulll:.
Then T is ~ - s t a b l e
< oo.
The clase of all/a-stable systems equipped with the induced norm, will be denoted by 15~T~"(t~). This class is s Banach space and in paxticulax it is a Banaeh algebra with multiplication defined as composition. In the cane where a = oo we refer to the space B~T~n(t ~176as the space of Bounded-Input-Bounded-Output (BIBO) stable, or simply, stable systems. Moreover, the following fact can be easily checked. nite rn • n block lower triangular matrices of the form
To(O) 0 ---/ r~(X) T~(O) 9
i
where Tk (i) are rn • n matrices for all i, k such that
,up I(n(k) n(k-
x) . . . n ( 0 ) ) l ~
<
II
3.2.2
oo.
T h e Small Gain T h e o r e m
We now present a sufficient condition for t~-stability known as the small gain theorem [10]. Let H,, H2 be/a-stable systems in feedbadc as in Figure 1. Also let uz E t~n and u2 E tram. Then T h e o r e m 3.1 The closed loop is ~ - s t a b l e (i.e., M1 signals in the loop lie in ~ o f appropriate dimensions) i f
IIH, II,~:.~-(,-) IIH211s~.~,.-(,..) < 1.
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
u,
-!
Y4
b. v
7
Z/1
H,
|
~a i
U2
H2
:
Figure 1" Feedback Loop 3.2.3
Exponential Stability
In the case when a system T is given in terms of a finite dimensional statespace description of the form
~(t
+
1) = A(t)=(t)
+
B(t)~(t)
9(t) = C(t)z(t) + D(t)u(t), t = 0, 1, 2 , . . . with (A(.), B(-), C(.), D(-)) being time varying matrices then Definition 3.3 The system T is exponentially stable ff there are constants c,, c2 > 0 such that for all to and z(to ) the following holds 1=(012 _< e, I=(to)12e'~('-'0)
t = to,
to + 1,
.
.
.
.
We also define the notion of stabilizability and detectability as follows Definition 3.4 The pa/r (A(.), B(.))/s called stabilizable if there ex/sts a
bo.od~d m~t,i~ r.,~tio, K(.) ~ . ~ t~.t t ~ ~y~t~m ~(~ + 1) = ( A ( t ) -
B(t)K(t))z(t)
is exponentially stable. Similarly, the pa/r (A(-), C(-)) /s detectable if there is a bounded matrix function L(.) such that the system
A finite dimensional system with (A(.), B(.)) stabilizable and (A(-), C(.)) detectable is called stabilizable.
3.3
T i m e (Shift) Invariance
An important subclass of the general time varying systems of the class s is the class of time (shift) invarisnt systems. If T 9 t~~ 'e ---* t~~ '" then
8
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
D e f i n i t i o n 3.5 T is time invariant if it commutes with the shift operator; J.e.~
Am T - TAn. The space of time invariant operators that are also ta-stable is denoted by ~ ~ ( g ~ ) w h e r e a - oo, 2.
4
Notational Convention
To avoid proliferation of notation we will often drop the m and n in the notation given in the previous sections when the dimension is not important or when it is clear from the context. Also, subscripts on the norms are dropped when there is no ambiguity.
5
Fundamental
Connections
In this section we provide the connections between the spaces of section 1.3 and B.rt(g a) where a -- oo, 2. We start with BIBO time invariant operators. The following fact can be easily checked:
Fact 5.1 Every element oftimxn (.A,nxn) detines an operator in l~T;n(~ =) via convolution (multiplication in the A-domain) and vice versa. Moreover,
)
omo phi ;
The above fact means that a tm~ operator T is associated with a sequence {T(i)}~0 E 11 which is the impulse response. This can be easily seen from the Toeplitz representation of T: T=
T(0) T(1)
I
9
0 T(0)
""/ .
:
The induced operator norm over t ~176 is exactly the tl-norm of {T(i)}~0. The sequence {T(i) }~0 convolves with the input sequence {u(k) }~~ E goo to produce the output sequence of T. In the space .A this translates to multiplication of the corresponding A-transforms. Finally, the isomorphy establishes that composition (of operators) in B~rt(t ~176translates to convolution in t I and, of course, multiplication in .A. Next we encounter the 12-stable operators. First, we have the following [11] concerning the linear bounded operators on s F a c t 5.2 An element R E s o o de~nes an operator from s to ~ m2 x l via multiplication. Moreover, any operator [rom s to s can be represented with some R E s the induced operator norm is exactly
JlRJJ~-m X R
CONTROLLERSFOR PERIODICAND MULTIRATESYSTEMS The bounded operators on the subspace 9t 2 of s next fact [11]:
9
are characterized by the
oo F a c t 5.3 An element X E 7~mx n defines an operator from "~.xt 2 to " ~ x l via multiplication and vice versa. Moreover, any operator from ~/2nxt to oo 7t~x 1 can be represented with some X E 7~rnxn; the induced operator norm is exactly IlXll,c=, 9
Having in mind that t ~ _ 7/~mxl it is not hard to establish that
F ~ t 5.4 Ao ~ a ~ o t io ~ ~ . d~. ~ open.to, io ~T~"(t ~) ~i~ ~ l tiplication in the A-domain and vice versa. Moreover, any operator in 7t~xn, B~T~n (f2) are
isometrical), isomorphic; i.e., oo
Again, the previous fact means that any function T(A) E 7~~176 defines an operator T in 8TZ(t 2) by multiplication and vice versa. The induced operator norm over t 2 is exactly J]T(~)]J
. T h e input sequence {u(k)}~~
E l ;~
ii ii transforms to a function ~(A) ~ ~2 that multiplies ~(A) to produce the transform Tu(A) E 7t 2 of the output sequence; then, by inverse tranform we obtain the output Tu E f2.
6
Important
Theorems
Here, we present without proofs some theorems from mathematical analysis [22,16,14] which play a central role in our development. 6.1
Duality
We start with the two Duality theorems which we use in section IV. Let X be some normed linear space and let X* denote its dual. That is, X" is the space of all bounded linear functionals r on X. X" is a normed linear space equipped with the induced norm; i.e., if r E X* then
]lrl] =
sup
zEBX
I(z,r)l
where (z, r) means the value of the functional r at z. The right annihilator (or orthogonal complement) space of a subspace S of X is defined as
s " : (~ ~ x " . ( , , ~) = 0 v~ ~ s}. Suppose that Q is a subspace in the dual space X" of X. Then the left annihilator of Q is defined as
10
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
Finally, we say t h a t r E X*, z e X are aligned if (z, r) = Ilxll I1,'11 9w e now state the two main theorems: T h e o r e m 6.1 Let z be an element in a real n o r m e d linear space X and let p denote its d/stance from the subspace S. Then p = inf ]lz - kll = max ( z , r) kes ~cs~, u~ll
where the m a x / m u m / s ach/ered for some ro in S • . M o r e o v e r , / f the infimum on the left is achieved for some ko E S, then ro is aligned with z - ko. T h e o r e m 6.2 Let S be a suhspace of a real n o r m e d / / n e a r space X . Let z" E X" be a d/stance p from S • . Then p=
rain I l z * - r * [ l =
r'E $~
sup (z,z ~
zEB$
where the m i n i m u m on the left is achieved for some r o E S "L 9 Moreover, is the s u p r e m u m on the right is achieved for some zo E B S then z" r o* , X o are aligned. -
-
In our development we use the second theorem although the first can be equally well. We also present a basic fact t h a t shows the relations between the spaces ,4~xn,tlmxn and e~ F a c t 6.1 Every linear ftmctionM on t ~ x n /8 representable uniquely in the ~ol'm N
oo
i = 1 j----1 k = 0
=
e
And H =
e
holds; hence, ( l ~ x n )* = .A~x . It can be also shown that where the linear functionals are defined as above. 9
0
9
-
x
With this fact in mind and the duality theorems we can turn distance problems in t I to distance problems in ,4" or in c~ 6.2
Projection
Theorem
As a corollary to the duality theorems of the previous section we present the Projection theorem when the space X happens to be a Hilbert space. In particular, we have: T h e o r e m 6.3 Let X be a ////bert space and M a closed subspace os X. Corresponding to any z E X there/s a unique mo E Air such that Ilz- moll < IIz- roll for all m e M . Furthermore a necessary a n d sufficient condition that mo be the unique m / n i m / z e r / s that z -- m , E M "L. This theorem is used in section V to solve a minimization problem in s
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
6.3
Nehari's
11
Theorem
Nehari's theorem solves an important distance problem in the space s This theorem will be used in section V to solve s 7g~176 problem. First, we define the Hankel operator assgcisted with a function R E s Definition 6.1 Let R E s and u E ~ 2 . Then the Hankel operator with symbol R is the map F R : Tg2 --* (Tg~)"L with FRu = H(~,)~Ru
R(~),,(~) e ~:'.
wh~,,, R,, i~ o,~ run,rio,,
The induced norm IIrRII of this operator measures the distance from an element R in s and the space 7~~176 T h e o r e m 6.4 Let R ~. s
Then
inf
XE'~
I I n - Xll~.. = IIrRII.
Moreover, there exists a Xo E 7g~176 that achieves the inlimum.
IH. PERIODIC SYSTEMS: LIFTING AND THE OPTIMAL DISTURBANCE REJECTION PROBLEM
Lifting and LTI equivalence In this section we present the idea of "lifting" a periodic system to create an equivalent time (shift) invariant system. The development follows closely [13]. Let f represent the input-output map from s to t~~ 'e of s linear causal time varying system. Definition 1.1 The map f / s N-periodic if and only f f it commutes with the N th power of the right shift; i.e.,
y(A,,)N = (A~)NI
9
The following facts found in [13] can be easily checked. Let f be Nperiodic and Wm represent the isomorphism
w. :~," - ~ ~ a = {a(0), a ( 1 ) , . . . } - .
a(O) .(1) W~(a)=
{
.
a ( N - 1)
a(N) .(N + 1) ,
.
a ( 2 N - 1)
,...}
12
PETROS G. VOULGARIS AND M U N T H E R A. DAHLEH
Define the map L as
L(I) = %Iw,7, where Wp is defined similarly to Win. Then L(I) represents a system with inputs in t~~ and outputs in t~~ e. Moreover, as shown in [13] L ( f ) is LTI and the following hold: Fact 1.1 Given a m-input p-output linear causal N-periodic system f , one can associate via the map L a unique causal (pN x raN) LTI system L ( f ) with a transfer matrix F(A). Conversely, any (pN x raN) transfer matrix ~'(A) with F(0) block lower triangular can be associated by L -1 with a unique m-input, p-output,/inear, causa/, N-periodic system. N o t e : In the above fact, F(0) block lower triangular means that F(0)=
Foo
0
0
Fl0.
Fll.
0.
0 0
FN~
FNN
FNO FNI
...
0
where each submatrix F O has dimension p x m. This is exactly what is meant when the term block lower triangular is used hereafter. Also, we will use the term "lifting~ to indicate the action of L to a system f (i.e., F is the lifted system or F is the lift of 1). Finally, to avoid proliferation of notation, we hereafter use W generically instead of W,~ where the subscript n specifies the dimension of the elements of the sequence that this isomorphism acts on.
Fact 1.2 L preserves the algebraic properties and the norm. /n particular, [or a = co or 2
sup
uEBt~
Ilf-II~:
=
sup
wEBt~
IIFwll~;~
Hence, f is In-stable ff and only ff F = L ( f ) is/a-stable. The following examples illustrate how the equivalent LTI system is obtained from the action of the lifting isomorphism L. E x a m p l e 1.1 Consider the SISO map f " u ~ y given as y(o)
.(o)
y(1) y(2) y(3)
b(1) a(2) b(3)
9
=
~
o
o
b(O) 0 a(1) a(O) b(2) b(1) ~
~
o
o
0 0 b(0)
0 0 0
...
.(o)
u(1) u(2) u(3)
i
i
"
.
Since A2f = fA ~ it follows that f is 2-periodic. By lifting we create =w.={
.(1)
'
.(3)
"" "}'
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
,_
y(1)'
13
y(3)'"'}"
The resulting map F = L ( f ) = W f W - t 9fi ---, ~ can then be represented as
~(1)
_
F(1)
F(0)
~(1)
.
~
"
9
where
F(k) :
(.(2k)
a(2k-- 1) ) b(2k)
b(2k + l)
and
!
F(0)=
, k-
.
1,2,...
(o,0, 0) b(1) b(0) "
Clearly, F is a 2 • 2 shift invariant system (LTI) with P()~) = ~~1760 F(k))t k. Moreover, F(0) = F(0) is a lower triangular matrix in accordance with Fact 1.1. The previous example captures the main idea of lifting: all that lifting does is re-representing the original periodic system in a LTI fashion by grouping the elements of the input and output and creating a higher dimensional input-output pair; this should be clear from the matrix representation of a periodic system f. In the next example we give a state space description of the lifted system once we have a state space description of the periodic olle.
E x a m p l e 1.2 Consider a state space .description of a N-periodic map f given as
9(k + x) = a(k)~(k) + B(k).(k) y(~) = c ( k ) ~ ( k ) + D(k).(k) with A(.), B(-), C(.), D(.) N-periodic matrices (i.e., A(k + N) = A(k), k = 0, 1 , . . . etc.) It is easy to check that a state space description of F = L ( f ) is the following:
.~(k) = c~(k) + Da(k) where ~(k) = z(kN) and A = A ( N - X)A(N- 2)...A(0) B = (A(N- 1)A(N- 2)...A(1)B(0) C----
...
A(N- 1)B(N-
CO)A(o) U ( N - 1)A(/~ - 2 ) . . . A ( 0 )
2) B ( N - 1))
14
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
/
o,o, co)B(o)
o
D~
...
O(,)
9
.
C(N - 1)A(N -" 2)...A(1)B(0)
o /
o .
C ( N - 1)A(N - 2)...A(2)B(1)
D ( b / - 1)
Note that D is indeed block lower triangular. Also, note that the state dimension is the same with the periodic system; it is only the input and output dimensions that get higher. Suppose now that f is finite dimensional with a stabilizable and detectable state space description. In [3,19] it is shown that we can obtain a doubly eoprime factorization (def) of f by obtaining a dcf of the lifted system F. The key observation in [3] is that the factors of F obtained using the standard formulas in [11] possess the property of being block lower triangular at A = 0. To view this, let F have the state space description F ~ (A, B, C, D). Note that since F is the lift of f then the feedforward term D is block lower triangular. Then the eoprime factorization suggested in [11] is F = NtD[ "1 = D~*Nr with
-N,
,,)(o, ,,) D,
Nt Xa
=I
where Nt ~ (AK, B, C x , D), Mz ~ (AK, B, K, I) N, ,~ ( A u , B u , C, D), M, ~, (AM, M, C, I)
x, ~ (Ax, - M , Cx, Z), Y, ~ (Ax, B , - M , K, O) o
X , ... ( A M , - - B M , K, I), Y, -- ( A M , - M , K, 0) with K, M selected such that AK = A + B K , AM = A + M C are stable (eigenvalues in the open unit disk) and BM = B + M D , CK = C + D K . The above formulas indicate that the eoprime factors of F have as feedforward terms the matrices D or I or 0 which are block lower triangular. Hence, since L is an isomorphism, we can obtain a def of f, the factors being the images of the inverse map of the lifted LTI factors of F and therefore N-periodic. Also assuming well-p(medne~ [11,9] we can characterize all stabilizing controllers c in terms of these factors. In summary we have: Fact 1.3 Let F = Nz D~ 1 = DT, N, and
-N,
,,)(o, ,,) 19,
N, X,
=I
represent a dcf of F where the factors are given as in [11]. Then, the following represent a def of f : f = ntd~ x = d~lnr
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
(..
-he
w/~er~ na = L-X(/Cz), d~ -
dr
nl
15
.),
Zl
L - X ( D ~ ) , xa = L - X ( X z ) , ya = L - x ( 1 ~ ) , n , =
L-X(Nr), dr = L-X(Dr), zr = L-X(Xr), Yr = L-~(Yr) are N-periodic in B T v ( ~ ) for a = co, 2. Moreover, all ~-stabilizing time varying controllers
c or I areg / t e n by c
= (Yl -- dlq)(zz - ntq) -1 = (z, - qnr)-x(yr - qdr)
where q E Dry ( ~ ) . Note that in the above fact it is easy to check that q is N-periodic if and only if c is N periodic. Finally, along the lines of [24], it is shown in [3] that the optimal performance in periodic plants is achieved with periodic controllers, namely: Fact 1.4 Let Hp, Up, Vp be N -periodic and ~ - s t a b l e causal linear opera. tom with a = oo, 2. Then iaf
qe e.rv(t- )
IIH, - UpQg, ll~(z.)
- inf q , IIH~ - U p Q ~ V ~ l l ~ ( ~ . )
where Qp is N-per/od/c and in B,rv(ta).
2
The
D i s t u r b a n c e Rejection P r o b l e m
The standard block diagram for the disturbance rejection problem is depicted in Figure 2. In this figure, Pv denotes some fixed linear causal N-periodic plant, C~ denotes a time varying compensator (not necessarily periodic), and the signals w, z, I/, and u are defined as follows: w, exogenous disturbance; z, signals to be regulated; y, measured plant output; and u, control inputs to the plant. Pv can be thought as a four block matrix each block being a linear causal N-periodic system. The following common assumptions are made: A s s u m p t i o n 2.1 The system of Figure 2 is well-posed [11,9]. A s s u m p t i o n 2.2 Pv/s finite d/mensiona/and stabilizable. Assumption 2.2 means that P~ has the state space description Pp'(A('),(BI(')B2(')),
C2(')
'
Dx2(')
Dx2(') / D~2(-) )
with all the matrices above being N-periodic and the pairs (A(-),B2(.)) and (A(-), C2(-)) being stabilizable and detectable respectively. Also, a sufficient condition for Assuption 2.1 to hold is that D22(-) = 0.
16
PETROS G. VOULGARIS AND M U N T H E R A. D A H L E H
W
Y
1/
Figure 2: Block Diagram for Disturbance Rejection. Now let Tzw represent the rmulting map from w to z for a given compensator C~. The first two problems are deterministic: w is assumed to be any t a disturbance and we are interested in minimizing the worst case fa norm of z. Namely, our objective can be stated as (OB3a): Find Cr such that the resulting closed loop system is stable and also the induced norm ]]Tz~ I] over tm for a = co, 2 is minimized. The third problem we want to solve is stochastic: we assume that w is a stationary zero mean Gaussian white noise with E[ww T] - I and we seek to minimize the average noise power in z. This is nothing else but the LQG problem. So our objective is stated as (OILILqG): Find Cr such that the resulting closed loop system is stable and also M-I
lim ( 1 / 2 M )
M-*oo
E
trace(E[z(k)zT(k)])
k-----M
is minimized. In the sequel we show that these problems can be turned into equivalent LTI problems. Towards this end, we lift each of the four blocks of Pv and let P denote the resulting LTI plant. Consider now the disturbance rejection problem for the system P and let T~w denote the map from @ - Ww to ;, - W z . Note that Assumption 2.1 guarantees the well-posedness of the
CONTROLLERSFOR PERIODICAND MULTIRATESYSTEMS
17
1 2 ) , and let the following represent a LTI problem. Let P = IPI1 P2x PP22
doubly coprime factorization of P22: P2~ = NID[ "I = D r lNr and
-N,
D,
Nr Xa
=I
where the factors are obtained by the formulas in [11] (which preserve the block lower triangular structure..) Then it is well known [29,8,28,9] that all the feasible maps are given as T . . = H - UQV where H, U, V 6/?~rl(t a) and Q 6 BTV(~). Moreover, H, U, V are determined by P. Now, the following lemma shows the aforementioned equivalence for the first two problems. L e m m a 2.1 The (OBJo) with a = ooor 2 / s equivalent to the problem (OPTa)" inf IIH- UQVIIs~,<~.) Qcsr,(z-) subject to Q(O) is block lower triangular. Morover, if Qo is the optima/solution to the above problem then the optimM compensator for the periodic plant is the inverse lift of Co = (Ya - D, Qo)(X, - NaQo)- x = (Xr - QoNr)- l ( y r "- QoDr).
Proof
We use Fact 1.3 and Fact 1.4 to obtain
infllT,~[[ C,
=
infllH p Q,
-
upQpvpl I
and Hp = L-X(H), Up = L-x(U), Vp = L-x(V). Now, by Facts l.l and 1.2 it follows that
where Qp is N-periodic,/~ infllHP Q.
-
~rPQ~II = q-L(q,) inf IIH--UQVII. |
For the third problem we know from the classical general LQG solution (for example [15]) that the optimal solution consists of solving a pair of periodic Riccati equations. Specifically, using the asymptotic solutions of the Riccati equations a state estimator and a feedback gain is obtained; then putting together these two elements yields the optimal controller. Now, from the results in [1,27] and by assumming that the pairs (A(.), B~ (.)), (A(-), Cx(.)) are stabilizable and detectable respectively we conclude that the asymptotic solution of the Riccatti equations are periodic and hence the optimal controller is periodic. In fact, the stabilizability and detectability assumptions can be further weakened ([2,27]). Therefore, it is enough to search over periodic controllers. But then by looking at the
18
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
lifted problem this is a standard L Q G problem in LTI systems. Moreover since E[Cv@ T] = I and M-1
trace(E[z(k)zT(Ir
lira (1/2M) Z
~r....co
k=-M M-1
trace(E[z(k)zT(Ir
( l / N ) limco(1/2M ) Z kf-M
then by solving the LTI problem subject to the causality conditions on the LTI controller will result to the optimal controller for the periodic system. But the LTI problem is inf
QET~--
IIH UQVII -
subject to Q(0) being block lower triangular. We should empasize that I I H - UQVII~2 is not an induced operator norm. It should be clear that what we mean is .llH(~)-//(~)Q(A)V(A)iI~ 2 . , With this in mind we have the following lemma. |B
L e m m a 2.2 The (OBJLQG) is equivalent to the problem (OPTLQG). inf l i H - UQVIIu2
QE'~"
subject to Q(O) is block lower triangular. Moreover, ff Qo is the optimal solution to the above problem, then the optimal compensator for the periodic plant is the inverse lift of Co = (Yt - DzQo)(Xz - NaQ.) -~ = (Xr - Q o N r ) - ' ( Y ~ -QoDr).
3
Section Summary
In this section we defined the problem of optimal disturbance rejection in periodic systems. In particular, we defined the /oo to t~~176 optimal performance objective (OBJoo), the t 2 to t 2 optimal performance objective (OBJ2), and the optimal LQG objective (OBJLqG). Via a lifting isomorphism [13] and the results of [3] we transformed these problems to equivalent LTI ones. These LTI problems involve an additional causality constraint on the feedforward term of the equivalent compensator which transforms to a constraint on the Youla parameter Q and makes the problem nonstandard. It is these problems that we approach in the following sections.
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
I V . O P T I M A L l ~176T O {oo D I S T U R B A N C E TION IN PERIODIC SYSTEMS
19
REJEC-
In this section we present the solution to the problem of optimal rejection of bounded persistent disturbances in periodic systems. In the previous section we defined precisely this problem i.e.,(OBJoo) and demonstrated that this problem transforms to an equivalent LTI problem in the lifted domain i.e.,(OPToo). Having in mind the characterizations of the various spaces introduced in section If, we can equivalently state (OPToo) as inf
QEI l
-
UQVll ,
subject to Q(0) being block lower triangular. The unconstrained problem i.e.,when Q(0) is not constrained to be block lower triangular, is solved in [5,6]. In [5,6] the problem is transformed to a tractable linear programming problem, via duality theory. In this section, we show that the same approach can be extended to yield the optimal solution for the constrained problem. In particular, we show that the causality constraits on Q(0) can be transformed as linear constraints on the closed loop m a p of interest i.e.,Txw. W e do this in two alternative ways. Once this is done, the duality theory can be used to provide the solution. Also in this section we consider, as an application, the problem of robust stabilization in periodic systems. W e indicate that this problem can be analyzed without introducing conservatism by considering the same problem for the equivalent LTI system. The latter is in fact a performance problem.
Problem Solution Clearly, if in (OPToo) we remove the constraint on Q(0) then the problem becomes the standard t I optimization problem [5,6]. In [5,6] the authors solve the problem by solving the dual problem with linear programming methods. W e can solve (OPToo) by extending the method in ~,6] to account for the constraint on Q(0). To show this, assume that U(A), I~'(A) have full row and column rank, respectively, for almost all A; we will come back to the general case later on. Also assume that U(A), V(A) have no zeros on the unit circle. Let now {P, },=x N, be as in [5,6] the basis for the functionals in c~ that annihilate the space S, = { K : K - U Q V , Q
i.e.,
(UQV, Pi) =
e tlmx,}
0 Vi = 1,2,...,N,,
Q e tlmx..
These functionals are attributed to the unstable zeros of U and V. Suppose now that we are able to find functionals {Xy }]=I in C~ having the
20
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
following property (PROP): if K E S, then
(K,X{) = 0
Yj = 1 , 2 , . . . , d
K = UQv
and
if and only if 1 :IQ E t m x.
with
Q(0)
block lower triangular.
Next, define S as S = { K : K = UQV, Q E lXmxn, Q(0)
block lower triangular}.
The following lemma, given without proof, stems from standard results in functional analysis (for example [16]): L e t n t n a 1.1 Let {Pn },=, N. E c~ as above and let {X/}~=z in C~ satisfy (PROP) as above. Then the annihilator subspace "LS of S can be
characterized as ~ s = , p . . ( { P ~ } . =No, u {x~}]=~) |
All that the above lemma says is that the functionale {Xj }f=x add the extra constraints of causality of Q(0) required to solve (OPToo) by enlarging the subspace • to • S . Since we now have a complete characterization of a. S we can proceed exactly as in [5,6] to solve (OPToo). Namely, using duality we can transform (OPToo) to a maximization problem inside B( a"S): By Fact 6.1 (Crux.) o 9 = tmx x . ; m o r ~ v e r if M is the subspace in c o x . defined as M = span({Pn }n=l ~V. t9 {Xj}y=I) then from the definitions of {Xj}y= x and {Pn} N~ it is easy to verify as in [5,6] that M • = S which implies that S is weak ~ closed. Hence inf I I n - KII =
KeS
rain I I n - KII = sup (G,H) KE(~S)~ UES(~S)
but since • S is finite dimensional sup (G,H)= uEs(xs)
max (G,H) uEs(~s)
therefore infllH-KIl=
K~S
max
a~B(~s)
(G,H).
The right-hand side of the above equality can be turned into a finite dimensional linear programming problem [5,6] and hence we obtain the optimal G = Go. The optimal/to is found by using the alignment conditions [5,6]
(Go, H -
K0) = IIH - K011.
CONTROLLERSFOR PERIODICAND MULTIRATESYSTEMS
21
In the sequel we show how to obtain these {Xj }[=t. First, we associate a functional Rj = {Rj(O), O, 0,...} in c o for every element j of Q(O) with indices (/j, mj) that has to equal 0 (i.e., the dements that are not in the block lower triangular portion of Q(O)) as follows: we pick Rj(O) to have the same dimension as Q(O) and also to have all its entries but one equal to 0. The nonzero entry is taken to equal 1 and its indices are precisely the ones that correspond to j (i.e., (lj, mj)). Hence, if r equals the number of elements in Q(O) that are necessarily zero then
(Q, Rj) = 0 Yj = 1 , 2 , . . . , r if and only if Q(0)
block lower triangular.
Therefore, (OPToo) can be rewritten as inf Q IIH - UQVII with QEt t,(Q,Rj)=O
Yj=l,2,...,r.
Next, we present the solution to (OPToo) by following two approaches.
1.1
Approach
1
By performing an inner outer factorization [11] for U, V we obtain U = ~iUo,
V=
Vo~
where the subscript i stands for "inner" and o for "outer"; i.e.,
Note that Ui(A), ~(A) are square matrices. Also, the various factors Ui, Uo, Vo, P~ may not in general correspond to a N-periodic system (i.e., at A = 0 they may not possess the block lower triangular structure). Our goal is to reflect the constraint on Q(0) to the product K -- UQV. Note that if U(0) and V(0) are square and invertible matrices then the situation is simple. To realize this, note that by Fact 1.3 both U(0), V(0) are block lower triangular. Hence, since K(0) = U(0)Q(0)V(0), we have that Q(0) is block lower triangular if and only if K(0) is block lower triangular. Therefore, the functionals we are looking for are simply the Rj's as defined in the beginning of the section. However this might not be the case: U(0), V(0) might not be square and/or they might not be invertible due to, for example, a delay. Because of this, we proceed by reflecting the constraints first on UoQVoand then on K. Towards this end let Z = UoQVo;the following propositions show how Z is affected due to the constraints on Q.
22
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
P r o p o s i t i o n 1.1 Let Z ~. t x then 3Q E t I
with
(Q, Rj) = O Vj = l , 2 , . . . , r
and
Z-
UoQVo
ff and only if
Z(0) E SA = {Uo(0)AVo(0) :A
block lower thangular matrix}.
Proof The "if" direction goes as follows: Let U~, Pot denote any right and left stable inverses of Uo, Vo respectively. Then U~., Pot E t x. Let A be a block lower triangular matrix such that Z(O) - Uo(O)AVo(O); define QA = {A, 0, 0,...} and let Z = Z - UoQAVo then Z E t 1 and Z(0) = 0. Define now Q as Q = UorZVoa + QA. It then follows that Q E t x, Q(0) is block lower triangular and Z = UoQVo. The "only if" direction is immediate. | Proposition 1.1 shows that only Z(0) is constrained to lie in a certain subspace (i.e., SA) otherwise Z can be arbitrary in t x. Hence, the problem of specifying constraints on Z in terms of functionals is finite-dimensional. Note that the characterization of this subspace is independent of the choice of right and left inverses for/.To, I/o respectively; hence it is exact. In the next proposition the annihilator subspace SB of SA is characterized and from this the annihilator subspace of Z is generated. P r o p o s i t i o n 1.2 Let {Bj }j=l j e be a bas/s for the finite-dimensional Euclidean space
sa
= {B.
uT(o)BV:(O) e
~p~.({Rj(0)}~=~)).
Define the following functionals in cO:
R~i={Bj,0,0,...} Vj=I,2,...,jB. Let Z E gI then
z(o) ~ sA if and only ff (z,R,j) = o vj = 1,2,...,jB. Proof Since the only nonzero element in Rzj is the first one, it is clear that the condition (Z, Rzi) - 0 is equivalent to the condition (Z(0), Bj) -- O where (e, .) is interpreted as an inner product in the finite-dimensional Euclidean space of matrices. Now, let A be a block lower triangular matrix and B E SA • then since
(Uo(O)aVo(O), B) =
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
23
it follows that B E SB. Conversely, it also follows that if B E SB then (Uo(O)AVo(O),B) = O. Hence, SAJ" = Sa or equivalently
SB• = SA
|
which proves the proposition. The computation of and is presented next.
{Bj }j--1 j8 for the general nonsquare case is routine o
}J~l
C o m p u t a t i o n o f {Bj Herein, we indicate how to find all matrices B (of appropriate dimensions) such that there are real numbers c~ with f.
uy(o)Bv.~(o) = ~ ~,R~(O). i--I
First, we consider the following problem (PA)" Let M be a given matrix and {Ei}/iffil be a given set of linearly independent matrices. Find all matrices W such that MW
=
~ ciE~ i
for some real numbers ci. The solution to (PA) goes as follows: i Let Ei be partitioned to column vectors as Ei = (e~, e ~9, . . . , en) and W = (wl, w 2 , . . . , w , ) Also, define Pk = - ( e ~ , e ~ , . . . , e ~ ) for k = 1 , 2 , . . . , n and cT = (Cl, c 2 , . . . c l ) . Then,
(M P1 P2 .
Define P =
then (0)c
and ~ =
pk)('~)~=o. wl w2 .
. Let
N
=
(diag(M, M,..., M) P)
= 0. Therefore, we can compute a basis for W by computing
basis for the null-space of the matrix N which completes the solution to (PA). In view of the above construction if C = BvT(o) then we can compute a basis for the space a
{Cj }~c__.1
r
s= = { c - u ~ ( o ) c = ~ ~,R,(o)}. i=1
24
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
Then we can compute a basis {Dj }jj o=: for the space
So
=
{D
"
=x)}. Vo(O)D E. span({Ci } i.ic
Now, by defining j 8 = j o and Bj = D T required basis for SB.
j = 1 , . . . , j B we obtain the
Since the matrices Uo(0), Vo(0) have full row and column rank respectively, it can be easily checked that the dimension of Sa is going to be at most equal to the dimension of span({Rj(0)}~=x) i.e., jB _< r. Therefore, Z will be "less constrained" than Q. In fact, there are cases as in Example 1.2, where Z turns out to be unconstrained in t 1. In this case, SB = {0} or equivalently SA is the whole finite-dimensional space of matrices (of appropriate dimensions) and hence, the solution to (OPToo) is the same to the standard unconstrained problem in [5,6]. The equality j s = r occurs when U and V are square. In this case, the computation of {Bj }~=I j B is immediate. Namely, Bj -- u'oT(O)Rj(O)V~'T(o)
j = 1 , . . . , r.
In view of the above proposition,, ( O P T = ) can be stated as
inf [In
-
Z
UeZV~
II
with get
x,(Z,R.,)=O
Yj=I,...,jB.
We now show how to obtain the funcfionala {Xj }]=x that have the property (PROP) mentioned in the beginning of this section. T h e o r e m 1.1 The functionals
Xj = UiRz~l~
j = 1,2,...jB
satis/y (PROP). Proof
Consider the bounded operators Tu,, Tv, on t 1 defined as g
(Tu, X)(,) =
-
I"----0 t
(Tv,X)(0 = ~_~X(r)I~(t - r) ~----0
where X e t I. Their weak" sdjoints ~ ,
~v~ on co which are given by
oo
(~,Y)(t) - E U~(r)Y(r + t) I"--0
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
25
oo
( '~v~Y)(O = Z Y ( ~ " + t ) v / T ( T ) Ir~.0
where Y E co Notice, that since U,, V/inner then 0~/(A-1)01(A) = I a n d ~ / ( ~ - I ) ~ / T ( ) ~ ) __ I. Note also that ~ , Tu~ represent multiplication from the right whereas ~v~, Tv~ represent multiplication from the left. Hence, it follows that i~u, Tu, I ~d ~,Tv, = Z Interpreting
U, Z V~ -- Tu , ( Tg, ( Z ) ) and
U~R. V~ = Tv , ( Tv~ ( R, ) ) with we can verify that
(u~z~, U~R,~) =
(Z, R,)
Hence, if X = Ui Rj V/then
(UQV, X) = 0 if and only if ( Z , R , ) = 0
|
which completes the proof.
Remark So far in this section we assumed that U(A), V(A) have full row and column rank respectively. However, there is no loss of generality since in the "bad" rank case (i.e., when the above assumption does not hold [5,6,17]) it is shown in [17] that in order to solve the unconstrained problem it is necessary to solve a square subproblem. In particular, we can partition
U, Vas
u=
u2 ,v=(9
where U, V are square and invertible. Let
K
v2) -
UQV
then
K21 K22 A necessary condition for the existence of a solution is t h a t / ~ interpolates U, V which is the aforementioned subproblem.
26
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
1.2
Approach
2
The solution presented so far required an inner-outer factorization for U, V. Next, an alternative approach which does not require inner-outer factorization is presented. For simplicity, we indicate the method in the case where V = I. The extension in the c a ~ where g ~ I should be immediate to the reader. Let U be written in the Smith form [12] as:
U=S1E(I
0).92
where S1,$2 correspond to square, stable transfer matrices with stable inverses and ~-,(~) = diag(Go(~), ~gl (~), . . . , At gt(~)) where G0(,~) is a diagonal stable transfer matrix with no zeros at A = O, and Ot ( A ) , . . . , Or(A) are scalar stable transfer functions also with no zeros at A = 0. Let (Bj }y=t j~ be a basis for the finite-dimensional space
(I)
=
B E ,pan({Ry(O)}~=l)}
o
T h e o r e m 1.2 Let T~sv~ be Che weak* adjoint operator on c ~ associated
with S~1 i.e., O0
(T~s7, Y ) ( t ) = E
S~ T (v)Y(r + t)
rm0
with Y E c~ Then the functionals
with s
- diag(GoT(O), A ~ l ( O ) , . . . , Ak~'I(0))J~j(A)
with 6 =
0, 0 , . . . }
for j - 1, 2 , . . . , J~B satisfy (PROP). ,oof
\ 0J stable right inverse of it. Hence, if Q = (I 0)S2Q then we can go through exactly the same arguments aa in Propositions 1.1 and 1.2 to show that if E t I then 3QEt I
with
(Q, R j ) = 0
Vj=I,2,...,r
and
if and only if (Q, Fy) = 0
Yj -- 1,2,...,j-~B.
-Q=(I
O)S2Q
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
27
Now, let F E span({Fy }~i) and define D E co as D(A) -- diag(GoT(O), AO~'I(0),..., AkO~"(0))fi'(A) then we can verify that as follows: --
--
Partition Q and F as Q -
(~'~,D) = (-~,F) Q0 ql
.
Fo and F =
fl
then
qk i=k
(r.~, D) = (do(A)Oo(A),doT(0)to(A)) + ~](~'0,(A)~,(A), ~'~7~(0)1,(~)) i=1
but Fo(A) - Fo(0) and fi(A) =
fi(O) Vi-
1, 2 , . . . , k hence
i--k
(s~, D) = (ao(0)Qo(0), GoT(o)Fo(o))+~(g,(o)q,(o), g, -'(0)f,(0)>= <~, F> As in the proof of Theorem 1.1, let
TsT~ be the operator
on t 1 associated
with the stable system Si-1 and let T~si., be its adjoint on c ~ Then by defining
x = ~ r , (D) we can show that Hence ($I ~Q, X) = 0 if and only if (Q, F) = 0 which completes the proof. The functionals Xy, j = I,...,J-5 obtained by the second method will contain finitelym a n y nonzero components. Namely, Xj(m) = 0 V m > k. This is so because Fj has all its elements but the first equal to zero and hence Dy(m) equals zero for m > k. In the first method however, the obtained functionals will not, in general, have this property. They will decay though, (since they lie in co) and the rate of decay will be dictated by the poles of Ui and ~ . Hence, it seems that the linear programming problem in the dual space [5] will be simpler when the second method is used, provided that a simple way to obtain a Smith decomposition of a transfer matrix exists.
28
1.3
PETROS G. VOULGARISAND MUNTHERA. DAHLEH
Examples
Here we show that in general (OPToo) has different solution than the unconstrained problem. Example 1.1 Consider an optimization problem in a 2-periodic singleinput single-output system with the equivalent LTI problem being as follows; inf IIS - vQII
"-A/
QEll
/ 1 0 / U(A)--/A such that Q(0) is lower triangular, where/t(,~) = 0 2 ' 0 1 We first solve the unconstrained problem following [5]" The basis for the functionals that annihilate S, = {UQ "Q E t 1} consists of the following: F~={
(10) 0 0
F 2 = { l 0o
The resulting optimal solution
0
..
01 ) , 0 ,
..
~. = H-
.}
U Q . is:
(1 0)
where @12,4)22 arbitrary in t I such that II~x211,, + 114,2211,, __< 1
and
Qu= /--~b21 2--~22 / -~21
2 - ~22
Also, I1+.11~, = 1.
Now, in the constrained case, we obtain by using the second approach the following extra functional: F 3 = { I 00 01 1 , I 00 011 ,0,...} The resulting optimal solution is: ~(~)=
I I .5~ 1 0 1.5
"
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
with
Q(~)=
Also,
(oo) o .5
ll
uP =
-++22 + ~22(0) 2 - ~2~
/
and let <~,,+,= H - UQ,+~, then
(
1 -+~x
~(2 - ~22(0)) -+~22 )
Hence, since 1l~2xllz, + II~b2211z, _< I we have that
I1~.~11~, >_ 2 thus I1+.~11~, > II+llt,. There are cases however, where the unconstrained problem and the (OPToo) are the same. This is indicated in the following example. E x a m p l e 1.2 Once again consider am optimization problem in a 2-periodic system with one input and one output and let the equivalent LTI problem be given as inf [[H - UQII QEl t
where H is 2 x 2, U m 2 x 4, Q m 4 x 2 with Q(0) block lower triangular. Suppose U = U,U. with U,(A) = A, 0o(A) = (0 I) then Uo(0) has full row rank. Finally, let Z = UoQ. It can be easily checked that Uo(0)Q(0) spans the whole 2 x 2-dimensional space of read matrices with Q(0) being block lower triangular; i.e., Q(0) of the form
0/
Q(o) =
l"
0
Hence, the problem is inf t gGL
IIH- UiZII
which is exactly the unconstrained problem.
29
30
PETROS G. VOULGARIS AND M U N T H E R A. DAHLEH
:! A o
~
:
o
l
I
|
I i
I |
:: |
]
C~
: .......................................
': !
~
Figure 3: Block Diagram for Robust Stabilization.
2
goo to goo Robust Stabilization of Periodic Systems
In this section we consider the problem of robust stabilization in periodic systems. In particular, we indicate that necessary and sufficient conditions for robust stability can be obtained by considering the same problem for the equivalent lifted system and using the results in [7,24,20]. More specifically, we encounter the situation depicted in Figure 3: We are given a nominal m-input, n-output, N-periodic, strictly causal plant Gp subject to additive unstructured perturbations. The perturbed plant is described as Gp + RpAe, where P~ is a known m-input r-output N-periodic causal time varying system, and Ae an unknown r-input n-output perturbation in a class D defined as : D = {A~: A~ E/~T~n(P ~ strictly causal with
[IAv[[< 1).
The objective is to find necessary and sufficient conditions for the existence of a single causal compensator C~ that stabilizes the system in Figure 3 for all A~ E D. Note that the strict causality conditions on Gp, Av exist exclusively to guarantee well-posedness of the problem [11,9]. Consider now the same problem for the lifted system and let G, R, A, C represent the lifts of Gp, Rp, A~, C~ respectively. Then, it is evident (see also Fact
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
31
1.2) that well-posedness as well as stability of the lifted system implies well-posedness as well as stability of the original system in Figure 3 and vice versa. Hence, it is enough to deal with the lifted system. A sufficient condition for stabilityfollows immediately from the application of the small gain theorem [9]. Namely, the system is stable is C stabilizes the nomnal plant G and satisfies
Iic(I-UC)-XRl[ _< x
.
In the case where C is LTI (i.e., Cv is periodic) and therefore C(Iin [7] that the small gain condition is necessary as well as sufficientfor stability when the perturbations are arbitrary time varying with norm bounded by 1; i.e.,these perturbations considered in [7] are not initiallyconstrained to be the image of a lifting isomorphism. However, as shown in [7] if the small gain condition does not hold then a strictly causal destabilizing A* E ffTNX"A'(tm~ Can be constructed such that the system becomes unstable. Now, the fact that A* is strictly causal trivially implies that AO is a legitimate perturbation for the lifted system; i.e.,A* corresponds to a A* in I). Therefore, the small gain condition is in fact necessary as well as sufficientfor the stability of the original periodic system given that we restrict ouselves to periodic compensation. If we allow any time varying controller C~ then C is not LTI necessarily. Note however, that it is the image of the liftingisomorphism. In this case, an absolute bound on the robustness of the system can be obtained by utilizing the results in [24,20]. Towards this end, let t~ = infc I I C ( ! - GC)-IRII where C stabilizes G and is in the image of lifting. Then using exactly the same arguments as in [24] we can show that there exists a robustly stabilizing controller C (being in the image of lifting) if and only if/J < 1. This implies that if p > 1 there exists a strictly causal destabilizing A ~ E /~TNx"lv (tm~ obviously A ~ corresponds to a Av~ in ~Dand hence the original periodic system is unstable. Therefore, if one wishes to design a stabilizing compensator C~ for the periodic system that maximizes the tolerance to time varying additive perturbations, the problem to solve is
GC)- 1R is LTI, it was shown
Cv
inf
stabilizing Gp
Ilc (i-G C )- ll
which can be transformed (by Fact 1.4) to inf
Q6t t
IIH- uQVil
with Q(0) block lower triangular and H, U, V stable time invariant operatom determined by G, R. This problem can be solved with the methods indicated in the previous sections. The bottom line of the discussion in this section is that periodic compensation for the periodic system can be used for m a x i m u m tolerance to unstructured additive perturbations. It should be noted, that the situation
32
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
does not change if we consider different type of perturbations (for example multiplicative): the small gain theorem gives necessary as well as sufficient conditions for stability. Moreover, the maximum tolerance can be obtained by periodic controllers that optimize a suitable performance objective.
3
Section
S ummary
In this section we presented the solution to the problem of optimal to ~ disturbance rejection in periodic systems. The key observation was that we can obtain a finite number of linear constraints (functionals) on the allowable subspace of K = UQV to account for the causality condition on Q(0) when considering the equivalent LTI problem. These functionals combined with the functionals of the unconstrained problem in [5,6] characterize exactly the allowable subspace for K. Therefore, to obtain the solution, the same method of solution for the unconstrained problem can be applied yielding a tractable linear programming problem in the same manner as in [5,6]. Also, we indicated that the problem of robust stabilization in periodic systems can be analyz~ed without introducing conservatism by considering the problem for the equivalent LTI system.
V. O P T I M A L S O L U T I O N S FOR T H E tn TO s D I S T U R B A N C E R E J E C T I O N A N D LQG P R O B L E M S IN PERIODIC SYSTEMS In this section we present the solution to the problem of optimal rejection of energy bounded disturbances i.e., (OBJ2) and also to the LQG problem i.e., (OBJLqG) in periodically varying systems. As we saw in section III both problems can be transformed to equivalent LTI problems i.e., (OPT2) and (OPTLqG). A compact way to present these problems is as inf I I H - uQVllw,,
QE~--
,~ = ~ , 2
subject to Q(0) is block lower triangular. The case ~ = oo corresponds to (OPTs) and the case/3 = 2 to (OPTLqG). The first problem (i.e., /3 -- oo) is a constrained 7t ~ problem. We solve this 7/~176 problem, by suitably modifying the standard Nehari's approach [11] in order to account for the additional causality constraint on the parameter Q. This modification yields to a finite dimensional convex optimization problem over a convex set that needs to be solved before applying the standard solution to the Nehari problem. The solution to the above convex finite dimensional problem can be obtained using standard programming techniques. Once this is done, we obtain the optimal LTI controller and, hence, the optimal N-periodic controller for the original N-periodic system, by solving a standard Nehari's problem.
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
33
The second problem (i.e., ~? = 2) is the problem of minimizing the 7/2 norm of the system in the lifted domain by using a compensator C'(A) with (~(0) a block lower triangular matrix. This corresponds to solving the LQG problem for the original periodic system. The solution to the latter problem is known and is obtained by solving a pair of periodic Pdccati equations. The solution we present here for the 7i 2 problem in the lifted domain is an alternative way of solving the LQG problem for periodic systems. This alternative solution involves only LTI systems and is obtained by utilizing the Projection theorem in Hilbert spaces.
P r o b l e m Solution First, we transform both problems to easier ones following, in essense, Approach 1 of section IV. Namely, by performing an inner outer factorization [11] for U, V we obtain u = U~Uo,
V = VoV~
where the subscript i stands for "inner" and o for "outer"; i.e., ~-T(~-l)~ri(A) = I and ~ ( ~ - 1 ) ~,~.(A) = I. We will also make the simplifying technical assumption that U(~), i~'(A) do not lose rank on the unit circle and hence Uo, Vo have stable fight and left inverses respectively. Note that the various factors in the inner outer factorization do not possess necessarily the block lower triangular structure at A - 0. Then we proceed by reflecting the causality constraints of Q(0) on UoQVo. Towards this end let Z = UoQVo; the following proposition shows how Z is affected due to the constraints on
Q. P r o p o s i t i o n I . I Let Z E 7i ~176 then 3Q 6. "Itc~ with
Q(O)
block lower triangular and
Z = UoQVo
if and only if" Z(0) E 5A - {Uo(O)AVo(O):A
block lower triangular matrix}.
The proof of the above proposition is exactly the same as in Proposition 1.1 where t 1 is replaced by 7i ~176Proposition 1.1 shows that only Z(0) is constrained to lie in a certain finite dimensional subspace (i.e., SA) otherwise Z can be arbitrary in 7t ~176One can easily find a basis for this subspace by considering SA • and finding a basis for this subspace. This is done in Chapter 5 and we briefly repeat the procedure here: For each element j of
34
PETROS G. VOULGARISAND MUNTHERA. DAHLEH
Q(O) with indices (li , mi) that has to equal 0 (i.e., the elements that are not in the block lower triangular portion of Q(O)) we associate a matrix R i with the same dimension as Q(O) that has all its entries but one equal to O. The nonzero entry is taken to equal 1 and its indices are precisely the ones that correspond to j i.e., (lj, m i). If r is the number of the elements in Q(O) that are necessarily equal to 0 then the annihilator subspace SA ~ is
= { B : tror(O)BVr(o) A basis {B1, B 2 . . . , Bin } for this subspace can be found in n routine way (see Chapter 5 for details.) When Uo, V, are square the computation of the basis is immediate. Namely,
nj --/.)roT(0)RjVo -T(0)
j ---- 1 , . . . r .
Now, if (e, e) denotes the inner product in the finite dimensional Euclidean space of matrices then Z(0) E .,cA if and only if (Z(O),Bj)= 0
Yj = 1 , . . . j v .
In view of the above proposition the optimization problem becomes 9
inf
gE'H=*,g(O)ES~t
IIH - U~ZV, IIn'"
We are now ready to approach the problems for each case/3 = oo and/3 = 2 separately.
1.1
T h e 7~ ~176 problem
We solve this problem by modifying the standard Nehari's approach [11] in order to account for the additional causality constraint on the compensator. This modification yields a finite dimensional convex optimization problem over a convex closed set that needs to be solved before applying the standard solution to the Nehari problem. First we assume that Ui, ~ are square. We will come back to the general 4-block problem later. The solution to the 1-block case is as follows: Let R - U~*H ~ * where ~)'/*(~) = ~TT (~-1), ~/. (~) __. ~ (~--1) and define for each J E SA the system R~ as:
ka( ) =
s).
For each J E SA, let F R~ represent the Hankel operator [11] with symbol R~. Before we present the solution we will need to compute [[FRsll using state space formulae. In particular we are going to compute the controllability and observability grammians [11] associated with FRs. Towards this end let R correspond via the Fourier transform to the double-sided (since R is not necessarily causal) sequence (R(i))~=_oo then R~ will correspond to
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
35
(Rs(i))~_oo with R](i) = R(i+ 1) Vi r - 1 and R j ( - 1 ) = R ( 0 ) - J. Let now G represent the stable (causal) system associated with the pulse response {0, R ( - 1 ) , R ( - 2 ) , . . . } and let (A, B, C, 0) be a minimal state space description of it. Let also G represent the stable system associated with the pulse response {0, R ] ( - 1 ) , R ] ( - 2 ) , . . . } i.e., G is the anticausal part of R$ but viewed as a causal (one-sided) system. Then it easy to check that G has the state space description (A, B, C, 0) with
A=
,
B=
(o) I
c=(c
. = - ,
'
g
i)
..=.
where J = R ( 0 ) - J. Finally, let We, Wo be the controllability and observability grammians for G i.e.,
wo
=
a
BB
k=O O0
Wo = Z ( A T ) k c T C A
k.
k=0
Then Wc and Wo are the solutions to the Lyapunov equations 9
W c - AWcA T = B B T ,
W o - ATWoA = c T c .
m
Similarly, let We, Wo be the controllability and observability grammians for G. Following [11] we have that ]]rRsll -- pl/2t~v,1/2Tz r ~x/2~j. Using the state ~,,c --o,,c
space description w e c o m p u t e
wo=( ~176 ..~T7
Wo :
-~- K
where
K= Z(AT)~CTCAk
(Wo --~T~
~7'Wo~
)
Note that K does not depend on 3. Also, since K ~ 0 then K = K l l 2 K 112 with K 1/~ ~ O. Now, proceeding with the computations and rearranging certain terms w e obtain:
= where
0
0 -J
+LTL
36
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
Hence
_
with H = J
(,0) 0 7
" The following lemma shows that Ilrn, ll
~onv~xin
Lemxna 1.1 ~ = infJesA [[I'RjI[ /s a
finite dimensional optimiz&tion of a convex and continuous functional on a convex closed set.
Proof
From the preceding discussion we have inf [[FR,[[= inf ~ [ ( M ~ ) ] .
JESA
-HES
where
S={
0
9
JGSA}.
J=R(O)+J,
Clearly, since SA is s subspaee then S is a convex set. Moreover ifH1, H2 e S, and t e [0, 1] we have
~[(M(tHI+(1-t)H~))] L
=
__.[(tM~I
)4-(
(1-t)MH~ (1-t)L)]
or
~ [ ( M ( t H l + ( 1 - t ) H a ) ) ] _ tV[(MLHI + _ M~2 L < )] (1 t)W[( )] which shows thst , [ ( ~"H)] is convex in H and consequently in J. Also, continuity o f , [ ( ~ ) ] claim is proved.
with respect to J is apparent and therefore our |
We are now ready to show how to obtain the optimal solution to ZG~',Z(O)r
Theorem 1.1 1. I ~ -
The following hold:
= inf ze~t-,z(o)cs~ [ I R -
Z}[~. -/a,
2. A minimizer Jo of the preceding convex programming problem of Lemma 1.1 a/ways exits. Moreover, if 2(, is the solution to the standard Nehari problem
inf IIRjo - X[I
XE~**
37
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
then the optimal solution Z, is given by 2o(A) = ,To + AFCo(A). Proof
For the first part note that since 0i(A), IT/(A), AI axe inner then IlU - u, z v , ll = IlU,'HV," - Zll = l i t -
Zll
w , imng 2(a) = z ( o ) + ag(~) with ~ arbitrary in ~t = we h~v~
Now, if J fi SA then from Nehari's theorem we have: inf I[Rs - Z[[ -- [[rrs[[ hence the first part of the proof follows. The second part of the theorem is immediate given that a bounded minimizer Jo of the convex minimization in Lemma 1.1 can be found in ..,CA (note SA is unbounded). In fact, this is always the case and the proof of it follows from the fact that the optimal solution Z has to be bounded: Clearly the selection Z - 0 is a legitimate one since Z(0) E SA. Hence /J~t- _< ]IH[[; if now [[ZII > 2 [[H[I then
I I H - u~z~]l > I[zll- IIHI] > IIHII > u~--. Therefore for Z to be a minimizer it is necessary that ilzi[ _< 2 I[HII and
hence ~[Z(O)] < 2 ][H[I which implies that the search for the optimal + can be constrained in a closed and bounded subset SA of SA. Namely,
sA = {J ~ sA"
~[J] < 2 Ilnll}-
But then the continuity of the cost implies that an optimal Jo can be found . = .
in S A which is bounded.
|
The previous theorem indicates what is the additional convex minimization problem that has to be solved in order to account for the causality constraint on Q(0). The following corollary is a direct consequence from the proof of the previous analysis. Corollary 1.1 The convex n~nimization problem of L e m m a
/J~t- -- ~ -
1.1 is
mi_n pl/2(W~/2WoW~/2) JES~
with W~=(W~ 0
0 I)'
and S = R(O) + S.
Wo=
(7~
~TT'
Wo
7 T - j J + ( _ ~ T w o _ ~ _~Two_~)'ATW~ II
38
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
The above convex programming problem can be solved with descent algorithms. In [26] the authors treating a problem of 7t ~ optimization with time domain constraints afire at a similar finite dimensional convex programming problem. As they indicate the cost might not be differentiable at all points and therefore methods of non-differentiable optimization are called for. Although these generalized descent methods might be slow they have guaranteed convergence properties. In [26] and the references therein alternatives are given to improve the convergence rate. E x a m p l e 1.1 Consider the equivalent performance problem in a 2-periodic SISO sytem of Example 1.1 where now we use the 7~~176 instead of the t 1. Namely, consider Pu-- =
inf
QE'~"
IIH- UQIIn-
subject to Q(0) being lower trangular, with H(A) --
(
)~ 0 - ~1
)
" FactoringU=UiUowehaveUi=
Let R = U~H =
0
01 0) 2 '
1 ' Uo=
( x/~ 0 ) theiR(0)= (00) , A = 0 , 0
2
0 2
0(~) =
(1 1) 0
B=(~
1
"
0)
SowecomputeWc =Wo= lwhichmeansthat C = ( 10) , D = O the unconstrained solution yields to p~.. -- 1. To compute the constrained solution let R1 = ( 00
BI=u~T(O)RI=
0
01 ) t h e n a b a s i s f o r S ~
consists o n l y o f
1 ) " Thus the allowable J's in the optimization of /
Corollary 1.1 are of the form J =
j2
-ja
"
The solution (via numerical algorithm) to the convex programming problem yields approximately
J~ =
(0 0
.75
and p~r
= 1.58
.
As expected, p~.. > p ~ . . The full 4-block problem i.e., when Ui and/or ~ are not square is treated analogously as in the standard Nehari approach [11] with the so-called 7iterations. In particular, using exactly the same arguments as in [11] the
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
39
same iterative procedure can be established where at each iteration step a 1-block (square) problem with the additional causality constraints on the free parameter Q needs to be solved. Hence, the aforementioned procedure of solving the 7/oo constraind problem is complete.
1.2
The
7"[2 p r o b l e m
The solution to this problem is obtained by utilizingthe Projection theorem as follows: Let again R = U ~ H ~ * and let Y = {Y(O),Y(1),Y(2),...} represent the projection of R onto 712 i.e.,Y = IIx2(R). Also, consider the finite dimensional Euclidean space E of real matrices with dimensions equal to those of Y(0) and let lisa represent the projection operator onto the subspace SA of E. Then
T h e o r e m 1.2 The optima/solution Zo for the problem #~2 -
inf [IH - U i Z ~ [[zo Z,E'I~,Z(O)ESA
iS given by Zo = {IIsx (Y(0)), Y(1), Y(2),...}. Proof The proof follows from a direct application of the Projection theorem in the Hilbert space s Let 7~s = {Z : Z E 912,Z(O) E SA} then 7~s is a closed subspace of s Also, let (., .) denote the inner product in s Viewing Ui and P~ as operators on s we have that Zo is the optimal solution if and only if
( H - U~Zo~, u ~ z ~ ) = o
vz E ~s
or equivalently (u; ~v~" - Zo, z ) = o
vz E ~s
or equivalently n~(u;~v~"
-
&) =
o.
But
II~s(U~llVi ~ - {IIs,~(Y(O)), Y(1), Y(2), . . .} fi ~ which completes the proof. I The above theorem states that only the first component of the classical solution Y is affected, The computation of IIsx (Y(0)) is routine; for example having an orthonormal basis (Bj }~=t for SA~ we have that f.
IIs~ (Y(O)) = Y ( 0 ) - Z ( Y ( O ) , Bj)Bj. ./-1
40
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
E x a m p l e 1.2 Consider the Example 1.1 where now we are interested in the 7t 2 norm rather than the 7t ~176Clearly, the unconstrained problem yields #~2 = 1 with
(o o),0,...}
Z~~ which requires Qou(O) =
0 2
0
2 ) which is not lower triangular. For the
constrained problem by considering the normalized basis element B1 =
0
1/~f2 ofSA~ we obtain
Zo(O) =
~ ( o ) - (~(o), #x) =
Hence the optimal solution is Qo = (
2
0
(o o) 0
1
0
,0,...}. The resultingcost
g2 to g2 Robustness in Periodic Systems
In this section we consider the problem of finding necessary and sufficient conditions for stability in the presence of unstructured perturbations in the feedback loop of periodic systems. We limit our analysis to additive perturbations, but the same results can be produced for other types of perturbations as well (for example multiplicative perturbations). In particular we encounter the following problem: Let Gp in Figure 4 be a periodic system in ~ v ( t 2) and Ap be an unknown stable periodic perturbation in the class Vp -- {ZX~ ~ ~ r v ( t = ) - IIZ~pll < 1), and W'p a periodic weight in 8TV(t2). The question is under what conditions the pair (Gp, Ap) under feedback results in a t 2 stable system. A .umcient condition is the small gain condition [9] i.e. IIGpll < 1. In fa~t as we will indicate in what follows, this is necessary ms well: Let G, A represent the lifts of Gp, Ap respectively and let A represent the delay operator (i.e. the shift operator.) If IIGll > x then IIAGII -- IIGll > 1 and hence there exists [23] a Ax E 7/~ with II~xll < x such that the pair (AG, AI) in feedback is unstable. But then the pair ( G , A ~ is unstable with A ~ = AxA; mo~eove~ ~in~e I1~~ = II~xll < 1 and also A ~ is strictly proper then by Fact 1.2 we conclude that if IIGpII > 1 there is a destabilizing A~ E/)p with A~ being the inverse lift of A ~ In the discussion above, Gp will depend in general on the particular controller C~ we pick for the nominal unperturbed system. Hence, if we pertain to periodic controllers (i.e., periodic
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
1 |
,
|
W ~
! !
, ,,,,
I
! |
:Gp !
.
P,
v
|
!
| I |
| | | | | o I
1
..
| | ! ~
.
,......................................., Figure 4: Robust Stabilization. maps Gp(C~)), then in order to maximize the tolerance to perturbations in the class D the problem we have to solve is inf
C , periodic
[[Gp(C,)[[
which is a performance problem in periodic systems that can be solved with the method indicated in the previous sections. At this point, we do not know whether a general time varying Cv can give better margins than a periodic with respect to this particular class ~Dp of periodic perturbations. However, if we enlarge the class of perturbations to include generally time varying ones as then we have the following necessary and sufficient condition: T h e o r e m 2.1 The following statements are equivalent:
(i) There exists a robustly stabilizing time varying controller with respect to perturbations in l)u. (ii) There ex/sts a robustly stabilizing periodic controller with respect to perturbations in ~)~.
OiO inf
C. periodic
IlGp(C )ll "< 1.
41
42
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
Proof The (ii)=~(i)is obvious. For (i)::~(ii)consider the lifted feedback system ( G ( C ) , A) where C is the lifted controller; note that G, C, A are not shift invariant, nevertheless, all should posess a block lower triangular feedforward term. To be more specific, consider for example the relation z - Gv; then, the operation of G on v can be represented as the convolution z ( t ) - ~~176 g(t, r)v(r). G has a block lower triangular feedforward term if g(t, 0) is block lower triangular for all t. The same interpretation is valid for A, C. Now following [25] this implies that infk [[GAk[[ _< 1 for, otherwise, a strictly proper A, and hence a legitimate A~ E l)v, exists that destabilizes the system. But then, following the same arguments as in theorem 4.1 of [24] we can check that there is a controller C1 - A-t~CA t~ for some large enough integer kl, such that ]]G(C~)[] _< 1. Note that Cx has a block lower triangular feedforward term since C does and therefore it corresponds to a causal (not necessarily periodic) C1~. But then from Fact 1.4 there is a periodic controller Cp such that [[Gp(Cp)[I < 1 and therefore Cp will robustly stabilize the system for all perturbations in ~ . The (ii)r follows from the analysis of periodic perturbations in ~)p. | Hence, if we wish to maximize the tolerance to perturbations in the class/)~ the problem we have to solve is
inf
C. periodic
[[Gp(C~)] I
which is a performance problem in periodic systems that can be solved with the method indicated in section 4. Finally, we should note that the exactly the same analysis holds when we encounter the robustness problem for multirate systems.
3
Section S u m m a r y
In this section we presented the solutions to the optimal s2 to 12 disturbance rejection problem (7/~176as well as the solution to the L Q G (7/2) problem in periodic systems using the lifting technique. Both problems involved a causality condition on the optimal LTI compensator when viewed in the lifted domain. The 7/~176 problem was solved using the Nehari's theorem whereas in the 7i2 problem the solution was obtained using the Projection theorem. In particular, the 7~~176problem was solved by modifying the standard Nehari's approach in order to account for the additional causality constraint on the compensator. This modification yielded a finite dimensional convex optimization problem over a convex set that needs to be solved before applying the standard solution to the Nehari problem. The solution to the above convex finite dimensional problem can be obtained easily using standard programming techniques. In the 7g2 case the solution was obtained from the optimal (standard) unconstrained problem by projecting only the feedforward term of the standard solution to the allowable
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
43
Yl
Ul
K1T
L1T
K,~T
LpT
Um v
Figure 5" General Multirate System. subspace. Finally, we indicated that the small gain condition is necessary as well as sufficient for 12-stability in periodic and multirate systems and that general time varying compensation offers no advantage over periodic as far as robust stability to time varying perturbations is concerned.
VI. M U L T I R A T E SYSTEMS" O P T I M A L D I S T U R B A N C E REJECTION AND ROBUST STABILITY In this section, we utilize the methods of solutions to the problem of optimal disturbance rejection in periodic systems that we presented in sections VI and V in order to solve the same problem in the case of multirate systems. The most general configuration of a multirate system is depicted in Figure 5. In this figure each of the input sequences {uj (k) }~~ enters the system at every Tu~ seconds which might be different from, say, Tuj+l which is the rate of the input sequence {u~ +1 (k) }~~ Also, {yi (k) }~~ leaves the system at a rate of Ty~ seconds which might be different from the rates of the rest of the output and/or input sequences. A typical example of a multirate system is the multirate sampled data system (MRSD) of Figure 6 where not all of the input and output sequences are held and sampled at the same rate. In the case where not all of the rates of the input and output sequences are rational multiples of each other i.e., the sequences are not sychronized the problem of designing the controller is hard. In the case however where the rates are sychronized (i.e., any ratio of rates is rational) then the synthesis problem becomes considerably easier. In particular, lifting techniques can be used to transform a multirate problem to a LTI problem as with the case of periodic systems. Again, this lifting induces a causality constraint on the lifted controller which can be handled exactly with the same manner as in the periodic case. In the subsequent sections we give the details of
44
PETROS G. V O U L G A R | S A N D M U N T H E R A. D A H L E H
Ul . ~ ~
Uc I
Ycl
. . . .
Yl v
L~T K~T Um
G0(,)
o
Uc,~
Yc~
-I!-iii-.......
!
w
LpT
K, n T
Figure 6: General Multirate Sampled Data System. how this is done.
Lifting and Shift Invariant Equivalence Throughout this section we will deal only with the sychronous case. This means that for the system of Figure 5 the rates of all the input and output sequences can be written as integer multiples of some common time period T. In particular we have Tu~ = KiT, j = 1 , . . . , m and Ty, = L~T, i = 1 , . . . , p where we assume that the integers (L1,..., Lp, K1,...K,~) are relatively prime. The period T is called the basic time period (BTP). We will call a multirate system causal if the value of yi(k) Vi = 1 , . . . , m does not depend on any uj(l) with I > kL~/Kj Vj = 1 , . . . , p . For the multirate system of Figure 5 let N denote the least common multiple of the relatively prime integers (Li, L 2 , . . . , Lp, K1, K 2 , . . . , Kin). Also let Pi - ( N / L , ) , i -- 1 , . . . , p , Mj -- ( N / Kj ), j - 1 , . . . , m and P ~ = 1 Pi, M - ~ j = l Mj. In the sequel we introduce s lifting isomorphism [18,21] similar to the one in the periodic case, that transforms them to causal "single rate" systems. m
1.1
Lifting in multirate
systems
Before we proceed we give the following definition: Defin[tion 1.1 Given the set of integers {My )~--I, (Pi}~'=x and a P x M matrix D partitioned as S~--"
I D11D12... D1ml .
.
.
.
.
.
.
CONTROLLERSFOR PERIODICAND MULTIRATESYSTEMS
45
where each Dij is a Pi x Mj matrix we say that D satisfies the (P i, Mj) causality conditions if (D,i)o~ = 0 when (a-l) N
P~
N < O; -(/3- 1)~-j
l
~=
Ul(0) U~(0)
{
9
I/1(0) WMU --_ {
u,n'(O)
.
,...}--.
Urn'(1)
I
ul(M1) .x(Mx + 1)
U l ( M 1 - 1) ,~(o)
~,(M,-
,
Ul(1) U~(1)
1)j
,
,...}.
u 1 ( 2 M 1 - 1)
~2(M2) um(2Mm - I)
Similarly define Wp. Let finally, f represent the multirate system of Figure 5 and define the map L(/) = WpfWM ~ Then L ( f ) represents a system with inputs in t~M'" and outputs in t~p 'e. Moreover, it is not hard to verify the following [21,18]: Fact 1.1 Given the m-input p-output causal multirate system f, one can associate via the map L a unique causal "single rate" (P x M) system F = L ( f ) in s i.e., F has the P x M block Toeplitz structure:
Fo(0)
F=L(f)=
0
0
FI(1) FI(0) 0 F2(2) F2(1) F2(0) 9
9
...
9
9
Conversely, any (P x M) causal system F in ~PxM with Fn(O) satisfying the (Pi, Mj) causality conditions for a/l n = O, 1 , . . . can be associated by L -1 with a unique m-input, p-output, causal muitirate system os form os Figure 5.
46
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
We will use the term "lifting" to indicate the action of L to a system f (i.e., F is the lifted system or F is the lift of f). Also, to avoid proliferation of notation, we hereafter use W generically instead of W, where the subscript n specifies the dimension of the elements of the sequence that this isomorphism acts on. The properties of the map L are analogous to the periodic case. Fact 1.2 L preserves the a/gebra/c properties and the norm. In particular, sup
uEBt~
[[fu[[~;. =
sup [[FwJlt~ wEBt~
Hence, f is input-output stable if and only if F = L ( f ) is stable. This fact also holds i{s ~176 is replaced with s 2. From this fact we have that if f " t ~ '~ ~ t~~ '~, g 9t~~ '~ --, ~ ' ~ are compatible causal multirate systems then
1. fg is causal a multirate system with L(fg) -- L(f)L(g). 2. Provided the inverse exists (I + f g ) - I is a causal multirate system with L((! + f g)-~) - (I + L(y)L(g)) -~ . Next we define a class of input-output maps that generalize the notion of N-periodicity and will provide a framework for an important class of multirate systems [18]. D e f i n i t i o n 1.2 The m-input p-output linear discrete multirate system f of Figure 5 is called (P i, Mj ) shift varying if it has the following property: a simultaneous shift of each input sequence uj by Mj with j - 1 , . . . , m shifts each output sequence Yi by Pi with i - 1 , . . . , p. For example, the system in Figure 6 with Gc being s causal LTI continuous plant is a (P i, Mj) shift varying system. Note that N-periodic systems belong in the class of (P i, Mj) shift varying systems with Pi = Mi = N Vi, j. Also note that not all multirate systems are (P i, Mj) shift variant systems: for example if Ge in Figure 6 is a time varying continuous plant then the discrete input-output map is not (Pi, Mj) shift varying. Finally, to avoid misunderstanding, we should emphasize that not all multirate systems correspond to a multirate sampled data system. In fact multirate sampled data systems comprise a small subclass of multirate systems. For the class of (P i, Mi) shift varying systems we can easily check [18] the following Fact 1.3 f is (Pi, Mj) shift varying if and only if F = L ( f ) is shift invariant (LTI) with F(A) satysfying the (Pi, Mi) causality cinditions at A = O. Moreover, i[ f " s176 --, s g :s _., s are (Pi,Rk), (R~,M/) shift varying systems respectively then fg is (Pi, Mi) shift varying with L ( f g) - L(f)L(g) and provided the inverse exists (I + f g)-I is (Mj, P i) shirt varying with L((I + f g ) - l ) _ (I + L(I)L(g)) -1 .
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
47
The example that follows demonstrates how one can obtain a shift invariant equivalent via lifting for the simple case of the multirate sampled d a t a system illustrated in Figure 7. E x a m p l e 1.1 Consider the SISO multirate sampled d a t a system of Figure 7 where the input sequence u = {u(k)}~~ runs at a rate of T and the output sequence y = {y(k)}~= o is obtained by sampling data rate of 2T. Suppose t h a t G, is a continuous LTI system with state space representation as follows:
yo(t) = cozy(t) + m ur Discretizing the system at T we obtain
ze((k + 1)T) - Aze(kT) + Buc(kT)
ye(kT) = Cze(kT)+ Due(kT) with A = eAcT, B = f[ eAcrdrBe, C = Ce, D = De. Noting that y(k) = yc(k(2r)) and u(k) = uc(kT) the mulirate m a p f " u --~ y can be represented as an infinite matrix:
y(0)
u(1) y(2) u(3)
=
D
0
0
0
0
0
0
0
CAB CA3B CASB
CB CA2B CA4B
D CAB CA3B
0 CB CA2B
0 D CAB
0 0 CB
0 0 D
0 0 0
9
.
.
.
.
.
,
.
9
.
,
,
.
.
.
,
...
u(1)
.(2) .(3) .(5) .(6) .
Clearly, A f = f A 2. where A represents the shift operator. In this case K -- 1, L = 2, N = LCM(1, 2) = 2, M = 2, P = 1. When we lift the multirate system all we do is create ~ = WMu and ~ = Wpy by regrouping the elements of u and y in a way such that the input o u t p u t map F" ~ ---, ~ is shift invariant. In particular
u(1)'
u(3)'"
"}"
48
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
8 ! ! !
o ! ! 8
~
! !
T
u,
-
I
,, 2T
a~
./-i
y
-I
I ! i I I
I ! | ! I
Figure 7: A SISO Multirate Sampled Data System. Then F can be represented as
~(0) ~(1) .0(2)
#(3) (o o)
(CAB CB)
(o o)
--
(o o)
(D O)
(0 O)
9
.
9
~
(o o) ...
fi(1) ~(2) fi(3)
o
9
9
.
9
(CAB CB) (D O) (0 O) (CASB CA4B) (CASB CA~B) (CAB CB) (D O)
(CAaB CA2B)
a(o)
(0 O)
Is is now evident that F is a shift invariant (LTI) I x 2 system. Moreover, the state space representation of F is readily obtained:
9(k + 1) = a2~(k) + (AB B)a(k) .0(k) = c~(k) + (D o)a(k) with z(k) = zc(k(2T)). Note that F(0) satisfies the (P, M) causality conditions which state that F(0) = (. 0) where 9 stands for a "don't care" element. The full blown details of how to compute the state space description of the equivalent LTI system for the general multirate sampled data system of Figure 6 can be found in [18]. 1.2
Controller
Parametrization
In this subsection we use the shift invariance equivalence to characterize the class of all stabilizing, multirate, causal, (M j, Pi) shift varying controllers
.
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
49
for any (P i, Mj) shift varying multirate system. As established in the previous section the equivalence between (P i, Mj) multirate systems and shift invariant (LTI) ones is strong in the sense that algebraic properties and also induced operator norms are preserved. Hence, any P-input M-output LTI controller C that stabilizes the lifted LTI plant G with C(0) satisfying the (Mj, Pi) causality conditions corresponds to a (unique) causal multirate (Mj, Pi) shift varying controller Cmr that stabilizes Gin,. Now, utilizing the controller parametrization given in [11] for G we can obtain all controllers that stabilize G in terms of the Bezout factors and the Youla free parameter Q. Since we are interested in causal multirate controlers we should restrict Q to provide C the feedforward term of which (i.e., C(0)) satisfies the (M j, Pi) causality conditions; for, otherwise C will not corespond to a causal Cmr. However, similarly to the periodic case, this additional constraint on ~7(0) transforms to the same condition on Q(0). In particular we have the following [19,21]: Fact 1.4 Let G - Nt D~"1 _ D~"1Nr and
_,,. o.) (~
x,)
=I
represent a dcf of G where the factors are g~ven as in [11]. Then all causal multirate (Mj, Pi) shift varying controllers C,~, are the inverse lift of C -- (Yz - D , Q ) ( X ,
-
N I Q ) - I --~ ( i t
-- QNr)-l(Yr
--
QDr)
where Q any stable LTI operator with Q(O) satysfying the (Mi, Pi) causality conditions. Once a controller C that satisfies the (M j, Pi) causality conditions is obtained then the multirate controller C, nr can be realized with the Serial In Parallel Out (SIPO) and Parallel In Serial Out (PISO) architectures as in [19].
1.3
Problem Definition
The problem of disturbance rejection in multirate systems is completely analogous to the periodic case. The situation is depicted in Figure 8. In this figure, Gmr denotes some fixed linear causal multirate plant, Cmr denotes a causal multirate compensator, and the signals w, z, y, and u are defined as follows: w, exogenous disturbance; z, signals to be regulated; y, measured plant output; and u, control inputs to the plant. Gmr can be thought as a four block matrix each block being a linear causal multirate system. In particular we assume
a,,,,. = (a',,,',. e#,.
50
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
J
i!
Gmr
Y
Figure 8" Block Diagram for Disturbance Rejection. with G~r , G~, G ~ , , G~r b e i n g ( P ~ , M ~ ) , (Pt,M~'), shift varying respectively. The following common assumptions are made:
(P~,M~'), (P~,M'j')
A s s u m p t i o n 1.1 The system of Figure 8 is well-posed [11,9]. A s s t u n p t i o n 1.2 Grog is tinite dimensional and stabilizable. Now let T,w represent the resulting map from w to z for a given compensator Cmr. Again as in the periodic case we will consider the following performance objectives (OBJa): Find Cmr such that the resulting closed loop system is stable and also the induced norm []Tzw[[ over ~ for ~ - oo, 2 is minimized. (OB.]LQG): Assuming that w is a stationary zero mean Gaussian white noise with E[ww T] - I, find Cmr such that the resulting closed loop system is stable and also M-1
lira (1/2M)
M-*co
Z
trace(E[z(k)zT(k)])
kf-M
is minimized. As in the periodic case these problems can be turned into equivalent LTI problems. Towards this end, we lift each of the four blocks of G,nr and let
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
51
G denote the resulting LTI plant. Consider now, the disturbance rejection problem for the system G and let Tzw denote the map from ~ = W w to ~. = W z . Then it is well known [29,8,28,9] that all the feasible maps are given as Tzw = H - U Q V where H,U, V E 8 T t ( ~ ) and Q E 8TV(ga). Moreover, H, U, V are determined by G. Finally, the results in [3] can be extended to show that optimal performance in a (P i, Mj) system can be achieved by a ( M j , P i ) compensator. Now, the following lemma shows the equivalent LTI problems that have to be solved in order to obtain the solution to the original multirate problem. The proof is straightforward and, therefore, is ommitted. L e m m a 1.1 The (OBJa) with a = ooor2 is equivalent to the problem (OPTs): inf [IH - UQVI[BT,(t. ) Q~BT Kto ) (OPTLQG):
inf IIS(H - v ' q V ) l l . ,
QE~*"
where S is the constant weight S -
diagXl2(sx,S2,...,Sp)
with Si
-
constraints. The reason why the weight S enters in the OPTLQG problem is because M-I
lira (1/2M) Z
M-*co
trace(E[z(k)zT(k)]) =
k=-M M-1
lim (1/2M) Z
M---*oo
trace(E[z(k)S~Y'T(k)])"
k--M
Therefore the weighted problem in the lifted domain will result in the same cost (unweighted) for the original problem. Once an optimal Qo for the equivalent problem is obtained then the resulting multirate controller is the inverse lift of Co = (Yt - DtQo)(Xt -- NtQo) -x = (X~ - QoNr)-X(Yr - QoD~)
where Y~,Ya,X~,Xt are coprime factors of G 22 obtained as in [11]. 1.4
Problem
Solution
As we have seen solar, all of the development is in parallel with the periodic case. The reason is because the (M~, P~) causality constraints on Q(0) are of similar nature. All that these conditions require is that certain elements
52
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
(well-specified) of the feedforward term Q(O) have to equal zero; but this was the case in periodic systems as well, with the only difference being the location of these zeros in Q(0). Hence it is apparent that exactly the same methods of solution carry through to the multirate case. We now very briefly demonstrate how the solution to the optimal disturbace problem goes.
1.4.1
e ~176 Optimal Disturbance Rejection
We can proceed exactly as in the N-periodic case by simply modifying the functionals Rj defined in section IV as follows: Let J be the number of elements in Q(0) that are by the (Mj~, P~) causality conditions necessarily zero. To each j E {1, 2 , . . . , J} we associate the indices of the element of Q(0) that is necessarily zero. Also, for each j E { 1 , 2 , . . . , J} consider a matrix N# with the same dimension as Q(0) that has all its entries but one equal to 0. The nonzero entry is taken to equal 1 and its indices are taken to be the ones that correspond to j in the Q(0) matrix. Define the functionals {Rj}~= a as
Rj(O) = Nj; Rj(k) = O,
k=1,2,...
It is now clear that the solution of the N-periodic case applies immediately. 1.4.2
t 2 Optimal Disturbance Rejection
Working completely analogously as in the periodic case we transform (OPT2) to inf [[H - U ZP II .. ze~-,z(o)r
where now
S.4 = {Uo(O)AVo(O) :A
satisfiesthe (Mj, Pi) causality conditions}.
The solution to the above problem follows exactly the one in section V. 1.4.3
LQG solution
Also this problem transforms to z e n - ,z ( o )e s~,
where H 1 - S H , Ui1 and Ulo are inner outer factors of S U and SAI -- {UIo(O)AVo(O) " A
satisfies the ( M j , Pi) causality conditions}.
Again this problem is solved as in section V.
CONTROLLERSFOR PERIODICAND MULTIRATESYSTEMS
~ ~ l l | | i 8 l l
53
Rmr
o o o o e o
i,m,mo,~o~lo,Emool
ooom
1
i t
mr
: e e I
emr
e e
l !
I
: l I
|
~
I e
: e e l i i
| t : I i e
; .......................................
Figure 9: Block Diagram for Robust Stabilization.
2
Application to Robust Multirate Design
In this section, we encounter the problem of stability robustness in multirate systems. The development parallels that in the periodic case. We start with the robustness to BIBO unstructured uncertainty. Let Gin,.represent a (P~, Mj) shift varying plant subject to additive unstructured multirate perturbations as in Figure 9. Let the perturbed plant be Gmr + RmrAmr where P ~ , is a known (Am, Mj) shift varying stable system and A,~, is an unknown s stable causal multirate system with induced norm over ~oo satisfying Ilz~,ll < 1. We stress that Amr is not assumed to be a (Pj, Am) shift varying system; it is viewed as an arbitrary stable causal multirate system. The problem of robust stabilization is to find a single causal multirate controller C,~, , not necessarily (Mj, P~) shift varying, that stabilizes the perturbed class for all A,~,. This problem can be treated analogously with the N-periodic case: Let G, R, C, A be the lifts of G , . , , / ~ , , C,,,~, A,~, reRpx~v (s with spectively. Note that G, R are LTI and A is an operator in ~'TV induced over ~oo norm IJA[[ < I where N = Z. N., P = Z, Pi. Again as in the periodic case, if C is LTI (i.e., C~ is (Mj, P~) shift varying) then using the results in [7] we can construct a strictly proper A ~ with B - r r [[A~ < 1 that destabilizes the lifted system if and only if the B.rv(~ ~176 norm satisfies [ [ C ( I - G C ) - z R I I > 1. This A ~ is trivially in the image of the lifting isomorphism and hence it corresponds to a causal multirate A ~ .
54
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
Since stability of the lifted system implies stability of the original one and vice versa, we conclude that the small gain condition is necessary as well as sufficient for the stability of the multirate system. If now we allow any causal multirate controller C~, then by utilizing the results of [24,20] we can find an absolute bound on the robustness of the system. In particular, we have as in the periodic case that if
inf
C stabilizing G
IIC(I-
GC)-IRUsTv(t,)>
1
with C being in the image of the lifting, then a destabilizing perturbation exists. Therefore, maximum protection to additive perturbations can be achieved by a (M j, P~) shift varying causal multirate controller that infimizes the above performance problem in the lifted system. Similar results hold for multiplicative perturbations. The picture is analogous to the periodic case for t 2 to t ~ bounded perturbations and hence we will not repeat it here.
3
Section Summary
In this section we presented the solution to the optimal disturbance rejection problem for the general case of sychronous multirate systems. It was demonstrated that that via the lifting technique of [18,21] the problem transforms to an equivalent shift invariant problem that involves an additional causality constraint on the feedforwatd term of the lifted controller. The methods of sections IV and V developed for the periodic case carry through essentially unaltered in the multirate problem. Hence, the ~oo, t~ optimal performance and the LQG problems can be solved as in the periodic case. Finally, the stability-robustness problems can be analyzed without conservatism in the lifted domain.
VII. CONCLUSIONS In this chapter we presented the solutions to the optimal ~oo to ~ and g2 to t 2 (or, 7/~176disturbance rejection problems as well as the solution to the LQG (7~2) problem in periodic systems using the lifting technique. All problems involved a causality condition on the optimal LTI compensator when viewed in the lifted domain. In the optimal goo to go~ disturbance rejection problem, the key observation was that we can obtain a finite number of linear constraints (functionals) to account for the causality condition on the Youla parameter when considering the equivalent LTI problem. These functionals combined with the functionals of the unconstrained problem can be used exactly as in the standard t I problem to yield a tractable linear programming problem. The 7t ~ problem was solved using the Nehari's theorem whereas in the 7t 2 problem the solution was obtained using the Projection theorem.
CONTROLLERS FOR PERIODIC AND MULTIRATE SYSTEMS
55
In particular, the 7~~176 problem was solved by modifying the standard Nehari's approach in order to account for the additional causality constraint on the compensator. This modification yielded a finite dimensional convex optimization problem over a convex set that needs to be solved before applying the standard solution to the Nehari problem. The solution to the above convex finite dimensional problem can be obtained easily using standard programming techniques. In the ~/2 case the solution was obtained from the optimal (standard) unconstrained problem by projecting only the feedforward term of the standard solution to the allowable subspaee. Also we demonstrated that exactly the same methods can be applied to solve the performance problems in the case of multirate sampled systems. Finally, we indicated that the problem of robust stabilization in periodic and multirate systems can be analyzed without introducing conservatism by considering the problem for the equivalent LTI system.
References [1] S. Bittanti, P. Colaneri and G. De Nicolao. "The difference periodic Riecati equation for the periodic prediction problem," IEEE Trans. A-C, Vol AC-33, 1988. [2] S. Bittanti, P. Colaneri and G. De Nicolao. "The algebraic Riccati equation for the discrete-time periodic prediction problem," Systems and Control Letters, Vol 14, 1990. [3] H. Chapellat, M. Dahleh and S.P Bhattacharyya. "Optimal Disturbance Rejection for Periodic Systems," Technical report No. 89-019, Texas A and M University, college Station, TX. [4] T. Chen and B. Francis. "7/2 optimal sampled data control," IEEE Trans. A-C, Vol AC-36, April 1991. [5] M.A. Dahleh and J.B. Pearson. "l 1 optimal feedback controllers for MIMO discrete-time systems," IEEE Trans. A-C, Vol AC-32, April 1987. [6] M.A. Dahleh and J.B. Pearson. "Optimal rejection of persistent disturbances, robust stability and mixed sensitivity minimization," IEEE Trans. Automat. Contr., Vol AC-33, pp. 722-731, August 1988. [7] M.A. Dahleh and Y. Ohta. "A Necessary and Sufficient Condition for Robust BIBO Stability," Systems aad Control Letters 11, 1988, pp.271275
[8]
C.A. Desoer, R-W. Liu, J. Murray and R. Saeks. "Feedback System Design: The Fractional Representation Approach to Analysis and Synthesis," IEEE Trans. on Automatic Control, vol. AC-25, no. 3, pp. 399-412, June 1980.
56
PETROS G. VOULGARIS AND MUNTHER A. DAHLEH
[9] C.A. Desoer and M. Vidyasagar. Feedback Systems: Input-Output Properties, 1975, Academic Press, Inc, N.Y. [10] G. Zames and R.R. Kallmem. "On spectral mappings, higher order circle criteria and periodicaly varying systems," IEEE Trans. A-C, Vol AC-15, February 1970. [11] B.A. Francis. A Course in Hoo Control Theory, Springer-Verlag, 1987. [12] T. Kailath. Linear Systems, Prentice-Hall, 1980. [13] P.P. Khargonekar, K. Poola and A. Tannenbaum, "Robust Control of Linear Time-Invariant Plants Using Periodic Compensation," IEEE Trans. on Automatic Control, vol. AC-30, no.11, pp. 1088-1096, November 1985. [14] E. Kreysig. "Introductory Functional Analysis with Applications," NY: John Wiley and Sons, Inc, 1978. [15] H. Kwakernaak and R. Sivan. "Linear Optimal Control Systems," John Wiley and Sons, NY, 1972. [16] D.G. Luenberger. Optimization by Vector Space Methods, New York: Wiley, 1969. [17] J.S. McDonald and J.B. Pearson. "Constrained optimal control using the t 1 norm", Rice University Technical Report No. 8823, July 1989. [18] D.G. Meyer. "A New Class of Shift-Varying Operators, their ShiftInvariant Equivalents, and Multi-Rate Digital Systems," IEEE Trans on Automatic Control, AC-35, pp. 429-433, April 1990. [19] D.G. Meyer. "A Parametrization of Stabilizing Controllers for Multirate Sampled-Data Systems," IEEE Trans on Automatic Control, AC-35, no. 2, pp. 233-236, February 1990. [20] M. Khammash and J.B. Pearson. "Robust Disturbance Rejection in g1Optimal Control Systems," Rice University Technical Report, No.8920, May 1989. [21] R. Ravi, P.P. Khargonekar, K.D. Minto and C.N. Nett. "Controller parametrization for time-varying multirate plants" IEEE Trans on Automatic Control, AC-35, no. 11, pp. 1259-1262, November 1990. [22] W. Rudin. "Functional Analysis," McGraw-Hill, Inc, t973. [23] J.C. Doyle. "Analysis of feedback systems with structured uncertainty," IEE Proceedings, Vol 129, no.6, pp.242-250, November 1982.
CONTROLLERSFOR PERIODICANDMULTIRATESYSTEMS
57
[24] J.S. Shamma and M.A. Dahleh. "Time varying vs. time invariant compensation for rejection of persistent bounded disturbances and robust stability," IEEE Trans. A-C, Vol AC-36, July 1991. [25] J.S. Shamma. "The necessity of the small gain theorem for time varying and nonlinear systems," IEEE Trans. A-C, Vol AC-36, no. 10, October 1991. [26] A. Sideris and H. Rotstein ,,7/oo optimization with time domain constraints over a finite horizon," Proceedings of the ~gth CDC, Honolulu, Hawaii, December 1990. [27] C.E. de Souza, M.R. Grevers and G.C. Goodwin. "Riccati equations in optimal filtering of non stabilizable systems having singualr transition matrices," IEEE Trans. A-C, Vol AC-31, 1986. [28] M. Vidyasagar. Control Systems Synthesis: A Factorization Approach, MIT press, 1985. [29] D.C. Youla, H.A. Jabr, and J.J. Bongiorno. "Modern Wiener-Hopf design of optimal controllers-part 2: The multivariable case," IEEETrans. A-C, Vol. AC-21, June 1976.
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Discrete-Time Robust Adaptive Control Systems Miloje S. Radenkovic Department of Electrical Engineering University of Colorado at Denver Denver, CO 80217-3364
Anthony N. Michel Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556
Io
INTRODUCTION
The robust adaptive control problem has been the topic of numerous studies, publications and discussions over the past decade. Initially, it was observed that algorithms designed for the case of perfect system models, can not provide global stability of the adaptive system in the presence of unmodeled dynamics and external disturbances. In order to guarantee stability, a variety of modifications of the algorithms originally designed for the ideal ease have been proposed, such as or-modification, relative dead zone, signal normalization, projection of the parameter estimates and the like. In the context of deterministic adaptive control, many interesting results were obtained. Some of these results are concerned with the local stability of the adaptive system and with the explanation of the stability and instability mechanisms in adaptive algorithms. Important results which address the global stability of an adaptive system in the presence of unmodeled dynamics were also established. An excellent unification of the existing robust deterministic adaptive control algorithms is developed in [1], where also most of the significant works in the area. of deterministic adaptive control are cited. In addition to the references cited in [1], we point to the work reported in [2-4], where the authors prove that the projection of the parameter estimates is in fact sufficient to guarantee global stability in the presence of small unmodeled dynamics. In [1-4] global stability is achieved under the assumption that the unmodeled dynamics is "sufficiently small". A unified approach to robust deterministic and stochastic adaptive control is presented in [5] where the CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
60
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
algorithms with the projection of the parameter estimates are considered. It is shown that small algorithms gain may produce unacceptably large input and output signals in the adaptive loop. In contrast to the existing results in the adaptive control literature, reference [5] proves: (i) admissible unmodeled dynamics do not depend on any algorithm design parameter, and (ii) intensity of tolerated unmodeled dynamics, measured by the corresponding H~176 is the same as in the non-adaptive robust control. In addition to the above, reference [5] provides the new approach for the exact mathematical analysis of the self-stabilization mechanism and burstings recovery phenomena, in robust adaptive systems. Let us mention that large burstings are possible during the adaptation process when a-modification or parameter projection is used. If the reference signal is persistently exciting, in [6,7] it is shown that global stabilization can be achieved without incorporating leakage, relative dead zone, parameter projection or any other modification of the estimation algorithm. In [6] deterministic adaptive control in the presence of model mismatch is elaborated, and it is shown that due to the external excitation, burstings almost do not exist. Stochastic counterparts of these results is given in [7]. In the present literature on robust adaptive control, only upper bounds on the system performance are established in terms of the size of unmodehd dynamics. These upper bounds include many generic constants with unclear size and nature. There is no precise indication of how well a resulting performance is close to the case of the best reduced order controller. In the present chapter the adaptive control of a deterministic discrete time system in the presence of unmodeled dynamics and external disturbances is considered. The contribution of the chapter includes: (i) A methodology for the global stability analysis of robust adaptive systems based on the normalized gradient estimation scheme is developed. (ii) Global stability of the deterministic discrete time system in the presence of multiplicative and additive system perturbations and external disturbances is established. The case when at high frequencies unmodeled dynamics and external disturbances are large, complex and unstructured is treated. (iii) The self-stabilization mechanisms which are inherent in adaptive control and the possible appearance of burstings, are characterized analytically. (iv) It is proved that small algorithm gains/z may result in unacceptably large input and output signals in closed-loop adaptive systems. (v) It is shown that the proposed adaptive controller acts as a frequency selective controller. Specifically, at low frequenceis the proposed algorithm
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
61
provides fast adaptation and large loop gains in order to achieve stability and desired performances of the closed-loop system. At high frequencies, the estimation algorithms guarantees slow adaptation and behaves as a nearly non-adaptive controller. The chapter is organized as follows: Section II defines notation and terminology. Section III presents a methodology for the global stability analysis of robust deterministic adaptive systems based on the normalized gradient estimation algorithm. The underlying concept is quite natural and is based on constructing a suitable Lyapunov function for the different periods of adaptation. The algorithm for robust deterministic adaptive control is presented in Section IV. By using the method developed in Section III, it is shown that the projection of the parameter estimates is sufficient to guarantee global stability of the adaptive system. The characteristics of the proposed control scheme are as follows: (i) The minimum phase assumption of the nominal system model is not necessary to achieve global stability, (ii) persistency exciting conditions are not required, and (iii) conditions for global stability are given in the frequency domain, which make the presented results more attractive for engineering practice. Robust adaptive control without burstings is presented in Section V.
II.
NOTATION
AND TERMINOLOGY
In the following text we will consider discrete-time systems with transfer functions in the complex variable z. A transfer function T(z) is stable if all its poles are strictly inside the unit circle Iz I = 1. The frequency response of T(z) is {T(eil): Ill _< r} where f is the frequency variable normalized with respect to the sampling frequency. Consider a discrete time function x : ~ + --~ ~. Define the seminorm t
I[ z(t)[Ix:=
{~--~,~t-lz(j)2}l/2, 0 < ,~ < 1
(1)
j=l
where the superscript t is omitted when t = oo. When II ~(t)I1~ exists for all finite t, z is said to be in l~e. When ]] z(t) ]Ix is uniformly bounded over all t > O, z is said to be in l~. In our analysis by n~(t) we will denote t
n~(t) =
~
,~'-J z(j) 2, 0 < )~ < 1
(2)
j=l
H ~176 denotes the space of transfer functions T(z) which are analytic and bounded outside and on the unit circle in the z plane. S ~ is the operator
62
MILOJE S, RADENKOVIC AND ANTHONY N. MICHEL
defined
SXT(z) = T(~ll~z),
(3)
for a fixed A, 0 < A < 1. S x H ~ is the space of transfer functions T(z) such that SAT(z) E g ~. In other words T(z) E S x g r162if T(z) is analytic and bounded outside and on the circle Iz] = A1/~ in the z plane. For T(z) E H ~, the H Qr norm is defined by
il T(z)I1--'=
(4)
max IT(z)l I~1=1
Note that !1 T(z) IIM** is the operator norm induced by the choice of the l~[0, oo) norm on the input and output signals of the system T(z). Likewise, the norm of the SXH ~176 space is defined by
II T(z) I1~=:=11SXT(z) IIM**=I~1=1 maxlT()~l/Zz)[
(5)
This norm is induced by the 12x norm of the input and output signals of
T(z).
III.
ROBUST THEOREM
ULTIMATE
BOUNDEDNESS
In the present section we will define a time-interval as being an integer valued set assuming one of the following forms
[~i,t~] [~i, Zi )
A
=
{ h i , h i + 1,...,vii + n } , n >_ 0
ix =
{oti, oti-]-
( o,i, Zi]
(~i,Z,)
A =
1,...,hi+n-
1},n> 1
n}, n >_ 1
{ a i -I" 1 , . . . , a i
-I-
{ ~ i -t- 1 , . . . , ~ i
+ n --
1},n >_ 2
where ai > 1, hi E H, and /3/ = a i + n , n E A/'. When using the term interval (without modifier "time"), we will have in mind the usual definition of interval on the real numbers. Throughout the text when performing some majorizations, by ~i(t), i = 1,2, 3 , . . . , we will denote the nonegative functions which allow for the effect of the nonzero initial conditions. The function ~i(t) is given by ~i(t) = coiA~i, where 0 < coi< oo and 0 < A~i < 1. Let us consider the following recursion --t- 1) ~ -
,
;,,
n~(t)
+
2~le(t +
1)l-I-r(t)!
n~(t)
for t > 1,0 < / J < 2, with all variables having finite initial conditions.
(6)
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
63
Regarding relation (6) we introduce the following assumptions" (A1) V(t) 6 ~+ and V(t) < do < co, for all t > 1. (A2) "Exponentially growing rule" - the sequence n,(t) E ~+ can not grow faster than exponentially, i.e., for all t _> 1
n,(t) < Con,(t - 1) + ko + ~l(t), 1 < Co < co, 0 < ko < co
(7)
where n,(t) is defined in terms of e(t), i.e., t
.o(~)- ~ ~'-J~(j + 1) ~
(8)
j=l
for a fixed 0 < A < 1. (A3) nc~(t) E ~+ and for t > 1 (i) n#(t) > An#(t- 1), no(I) r 0, and n#(0) > 0, 0 < A < 1 (ii) no(t) < Cr ne(t)+ C#~ + ~2(t), 0 < C#1, Cr < co
(9) (10)
(A4) Sequence 7(1) E @~and depends on the sequence e(t), i.e., A'-JT(j) 2
< C~
j=l
A'-Je(j + 1) 2
+ k, + ~3(t) (11)
j=l
where 0 _< C.y,k~ < co, 0 < A < I.
(As) On the subsequence at, k = I, 2, 3,..., where S(a~ -4-I) > 0 and S(ak) <_ O, the following holds
s(~k + I)
_< cs
(12)
where t
sCt + ~) =~~~'-~
(13)
j=l
{(1 - 7P +
C,)e(j
+ 1
)2
-
2[e(j + 1)1 . {7(j)[}
a n d 0 < A < 1. Note that relation (6) describes the evolution of the Lyapunov type function V(t) for which we are interested in determining the l~ stability of the variable e(t). The term 7(t) tends to destabilize the convergence of the sequence V(t) and the 1~ stability of the sequence e(t). It turns out that the behavior of the sequence S(t) is crucial to the l~ stability of the sequence e(t). Throughout this chapter, the sequence S(t) will be referred
64
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
to as a bursting function. The reason for this terminology, as will be seen, can be found in the role played by the function S(t) in determining the stability (or instability) of the recursive scheme (6). As will be seen in the subsequent analysis, in the time intervals where S(t) <_ O, stability of the sequence e(t) follows trivially; however, in these time-intervals the Lyapunov function V(t) may be increasing (diverging), thus giving rise to bursting of the sequence e(t). As a consequence of this phenomenon, the function S(t) becomes positive, forcing the function V(t + 1 ) + ~n,(t) to converge by means of which e(t) is stabilized. The convergence properties of the recursion (6) will be formulated in the following theorem. T h e o r e m 3.1- Let the assumptions (A1) - (As) hold. If (A6) Pl = 1 - ~ - C , > O, then for all t > 1,
1 F+
~2C$2
C~,o + r
(14)
where L+I F
=
~'2Cd~2
c,~,
+ (z, + 1)c~k,
(15)
where L is a finite integer and 0 < e2 < 1, while other constants are defined in assumptions (At)-(A6). In (14) the positive constant ~0 is a consequence of the nonzero initial conditions and it is arbitrarily small for sufficiently large t. P r o o f : Let us first consider the case where the bursting function S(t + 1) > 0 for all t > 1. Using the definition of S(t) (Eq. (13)) and assumption (A3)(ii), from (6) one obtains
V(t + 1)+ S(t + 1) n,(t)
S(t) < v(t) + n4,(t- 1) -/~Pl
e(t + 1) 2 nr
(16)
where pl is given by assumption (A6). After summation from $ = 1 to N, from the above relation we obtain
~(t+ 1) 2
lim /~p~
S(1)
< V ( 1 ) + ,,~(0) < oo
(17)
~--1
wherefrom it follows that lim e(t + 1) 2
*-~
nr
=0
(~8)
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
65
Since assumption (A3)(ii) implies
n,(t) -< g'~' A l
(19)
from (18) it follows that lim e(t + l )~ ,-oo C~ 2 + c_~. max1 <_.~<_,e(r + 1) ~ = 0
(2o)
Suppose that there exists a subsequence {/k}, k = 1 , 2 , 3 , . . . , such that lim e(tk + 1) ~ = +oo.
(21)
m~x ~(~ + 1) ~ < ~(tk + 1) 2
(22)
k--oo
Then we have l
From (20) and (22) we conclude that lim
. . . .e(tk . . +. 1) . 2
c.~:(t k +
1)2
= o
(23)
i.e., lim e(tk + 1) ~ = 0
k---*oo
(24)
where we have used the fact that by assumption (Aa)(ii), C~2 ~ O. Eq. (24) contradicts the assumption (21), which implies that e(t + 1) is a bounded sequence. Then from Eq. (20), we derive that limt...~ e(~ + 1) ~ = 0, i.e., when the bursting function S(t) is positive for all t >_ 1, statement 1) of the theorem is correct. Let us now consider the case where the bursting function S(t + 1) _< 0 for all t > 1. From the definition of S(t) (Eq. (13)) we obtain
(1- ~I~ + CT) ne(t) < 2n,(t) x12 n.r(t) 112
(2.5)
where n,(t)is defined by Eq. (8) and nT(t)is defined by Eq. (2), where z(t) -7(t). From (25) and assumption (A4) we derive (1 - ~ - C-r)n,(t) < 2n,(t)xl~'[k. r + ~3(t)]
(26)
wherefrom we obtain
4k~
n,(t) _< p---~-+ ~4(t)
(27")
where Pl is defined by assumption (A6). Thus in the case when S(t + 1) < 0 for all t ___ 1, the statement of the theoerm is correct.
66
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
In the following we turn our attention to the case when there are intervals where the bursting function S(t) is positive and intervals where S(t) is nonpositive. Let us define the sequences rk and ak, k - 1, 2, 3 , . . . , as follows i
=
r;
< o"1 < 72 < 0,2 <
.
.
< "It < ffk < Tk+l < " ' ' ,
.
so that for k = 1 , 2 , 3 , . . . ,
S(t + 1) > 0 for t e [uk,rk+~), and S(t + 1) __<0 for ~ E Irk,ak) If S(2) > 0, we set r~ - 0 and r = 1. Then the interval [7"1,crx) is not under our considerations, since we are interested in the behavior of the S(t + 1) for t > 1. Since in time intervals [rk, uk), S(t + 1) _< 0, we conclude that relation (27) is valid for all t E [rk, u~). Let us analyze time intervals [ak, rk+l). Using the fact that for ~ E [uk, rk+l),S(t + 1) > 0, after summation from ~ = crk + 1 to N < rk+l, we obtain from (16)
S(N + 1) V(N+I)+
he(N)
S(ak + 1)
-
n,(ak)
N -~up,
e(t + 1) ~
Z
(29)
~=as,+l
Since by assumptions (A1) and (As) V(ak + 1 ) +
S(u~ + 1) n#(ak) <- do + Cs < oo
(30)
from (29) we obtain for N E [ak + 1, rk+l)
/v
e(t + 1) ~
PPl E
n,(t)
< do + Cs
(31)
tmaj,+l
Let us first consider the subsequence {p.}, n = 1, 2, 3,--., where lim._oo supn(rp.+l - a p . ) < Lip < oo if such a sequence e.xJsts. Using the "exponentially growing rule" (assumption (A2)), we obtain for t < rp. + 1 Lip-1
he(t) < C~"ne(cr,. - 1) + E
C~ [k, + ,fi (r - i)1
(32)
i=0
Since on the time intervals [rk, ak) relation (27) is valid, from (32) we derive
ne(t) <_ ~ Pi
+ l:e E C~ + ~5(t - L1p + l) i=o
(33)
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
67
Let us next derive a similar bound for ne(t) on the subsequence {k,,}, m = 1 , 2 , 3 , . . . , where l i r r ~ - o o ( r k . + l -~rk.,) = +oo. In this case, statement (31) implies that for arbitrarily small el > 0 e(t + 1)2 _<El n,(~)
(34)
for t E [ak. + 1, r k . + 1), except for a finite number of points tx < t2 < 9" < t L, where L =
pPlel J + 1
(35)
and tl > ak.. -I- 1,tL < rk,.+l. Denote by CL the finite set containing the points t ( , i - 1 , 2 , 3 , . . . , L and the point to - ak... Let us define the integers a(i), i - 1 , . . . , M < L as follows a(i) - a ( i - - 1 ) + D i - 1 , ~(0) - 0
where
l <_Di <_ L + I, i - 1 , . . . , M
(36)
(37)
< L
and N
Z Di
--
L+
(38)
1
i=0
Now we can define the points t~,(i)-i E CL, i -- 1, 2, 3 , . . . , M, and t~(i) E eL, i -- O, 1 , . . . , M , so that all points from the time intervals h
-
-
(39)
[]~a(i-- 1), tc~(i)- I] C CL
and all points from the intervals Ill = [ta(i)-I + I, ta(i)) ~. CL
(40)
It is obvious that tr
1 "--
t a c i - 1) + D i - 1 - 1, i -
1,...,
(41)
M < L
where D i _ l denotes the number of the points in the interval Ii, and Sa(O) = $0 -- O'k,,~, ~a(.~/)- 1 _~
"l'k,~+l
-- 1 and
ta(M) -
rk,,,+l.
(42)
It is obvious that in the intervals Ii, relation (34) is not valid, while in the intervals Ili, (34) is valid. From (34) and assumption (A3)(ii) it follows that for t E Ili,
max ta(i)_i
-{-~2(t)] (43)
68
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
where el is chosen so t h a t e2 - f,C*, 1-A < 1 F r o m the previous relation we obtain
"1" ~'1 [Cr
e ( r + i) ~ _< cIC~' ~e(ta(i)"l)
max t~
1 -
for all t E Ili, i -
-I- ~2(ta({)--1 )
(44)
e~
1 , . . . , M . Since 1
n,(t) < '1 _
A ,0r max<,-<, e(r + 1) 2 + A'-t~
(45)
from (44) we derive for t E Ili,
F ~'6(ta(i)-l) "] 1Cr - e,.
rle(t) 5 1 --1 ~2 rle(ta(i)-l)
(46)
where C63 =
e 2C,1,2
(47)
Cr t IC~x
and where we have used the fact t h a t e~ = 1-x 9 In order to evaluate e(t + 1) for t E h , we will use a s s u m p t i o n (A2). Thus we derive t-to(i-l)
,',o(~) < c~-'..,,-,,+' ,,,,(t<,_,)- 1) +
~
c~[k0 + ~,(t - ~:)] (4s)
k=O or
n,(t) < ~ , for t e h
(49)
where
~, = c2'-',,,(to(,_,)-
1)+ k,
Di-1-1
c ) + 6,(to(~_,>).
(50)
k=l
Let us find the recursive relation for/3i. From (46) and (49) it is not difficult to conclude
n,(t) < -
1 1 -
~i + e2
Cca + ~,6(to(i)- 1), t E I~i, i = 1, 2, 3 , . . . , M.
1 -
e~
(51)
and consequently, t h a t n~(t~(i_ I) - i) _< 1 - 1 -r ~ i - I "~" C~b._....~3 1 - e.~ .~_ r162( t a ( / _ 1 ) _ i )"
(52)
S u b s t i t u t i n g (52) into (50), we obtain
~, < .c~'-' ~,_, + c,~ -
l-e2
l-e2
Di-1-1
+ k, ~
k--O
c~ + ~8(t,,(,_,)_,)
(531
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
69
By induction, it is not difficult to see that i--1
Dj C~ ( 1 - 82) i - t if,
/~i
__~
i- t ~1 "}" E
(54)
n---t
(1 - r i-" C,3 + k0 ~ : ~ ~ - n - - 1
2-0
Csk
"~"(9(~a(1)-t)
for i - 2, 3,..., M _< L, where Bt is obtained from (50) for i = 1, i.e., Do-1
(55) 2=0
where we have used the fact that by definition to(0) = a2,.. After further majorizations, from (54), (55) and (27) we obtain for i = 1, 2, 3,..., M _< L /3/< p~(l
r
CoL+'
q" L ( i - r
C•3 + (L + 1)CsLko + 6,to(a2,~) (56)
where Cr is defined by Eq. (47). Finally relations (49), (51) and (56) establish the boundedness of ne(t) for $ 6 [ak,,, rk,,+l). In (56), ~t0(a2.,) is sufficiently small for enough large k,n. Thus the theorem is proved. O
IV@
ROBUST
ADAPTIVE
CONTROL
Let us consider the following discrete time SISO system with unmodelled dynamics
+
Ao(q-t)y(t + 1) = B0(q-1)[1 + At(q-t)Wl(q-t)]u(t) (57) Ao(q-1)A2(q-1)W2(q-1)u(t)+ A3(q-1)W3(q-1)w(t + 1)
where {y(t)}, {u(t)} and {w(t)) are output, input and external disturbances, respectively, while q-t represents the unit delay operator. Polynomials A0 (q- 1) and Be (q- 1) are given by
+ a,a q - h a
A0(q -1)
=
1 + alq -1 +
Bo(q -1)
=
bo + blq -1 + . . . + bnB
.
.
.
, (b0 # 0)
(.hs)
In the notation of the polynomials As(q - t ) and B0(q - t ) we use the subscript 0 to emphasize that the corresponding part of the system model (57) is structured parametrically, where by 0 we denote the parameter vector
0T - [al,..., a,~; bo, b l , . . . , b,8].
(59)
70
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
In (57) Ai(q-1)Wi(q-1), i = 1, 2, are multiplicative and additive perturbations. Transfer functions Ai(q -1) and Wi(q-1),i = 1,2,3 are causal and stable and Wi(q -1) are known. Regarding transfer functions Ai(q -1) and Wi(q-1),i = 1,2,3, we assume (St) A lower bound A0, 0 < )~0 < 1, for which transfer functions and
Wi(,~12z), i
= 1, 2, 3, are stable is known and
II zx,(=)I1~**~ 1, i -
1,2,3
where ,~ satisfies ,~1 + ,~0 < ,~ < 1, for ,~1 given by 0 < ,~1 < < ,~0. In (57) {w(t)} is the external disturbance satisfying (s~) I,~(t)l < k~, for all t >__ 1. The structure of the system model (57) is designed to capture the following frequency dependent nature of the a priory knowledge about the system. In the system description we consider the frequency interval 0 < f < ,-r. The same holds for -,7 < f < 0. (i) The system model is well known for frequencies ] where
O
implying that the transfer function ao(~) = Bo(~)/Ao(-.), (~ = e ~j, i 2 = - 1 )
(60)
is independent of 0. For 0 < f < fl
IW~(~x/"e~I)l <
7, i = 1, 2, 3
(61)
where ,~1 + ,~0 < ,~ < 1, 0 < ,~t < < ,~0 and 7 > 0 is a sufficiently small number compared with IC,(-)l. (ii) In the frequency range fx < f < f2, Ge(z) is highly dependant on O, but still [W,.(AXl2eil)[ __<7, for i - 1, 2, 3 where )i is the same parameter as in assumption (St) and Eq. (61). The dynamics in this frequency range may include parts of the known structure but unknown parameters such as resonant modes whose damping coefficient is unknown and the resonant frequency can vary between ]'1 and ]'2. (iii) In the frequency range f > f~, IW~(~/2e~l)l, i = 1,2, 3 is large and the system is dominated by those complex dynamics which are unknown in both structure and complemty. As a consequence of the pre~-ious description of the system models (57) we assume: ($3) There exists an unknown parameter set O ~ so that for every 6 E O', transfer function Be(z)/Ao(z) well matches the dynamics of the physical system at low frequencies f < ]'2.
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
71
From the previous system model description it is also obvious that at high frequencies f > f2, large unstructured and complex external disturbances are possible. Classical non-adaptive controllers can produce a large loop gain and achieve satisfactory performances at frequencies f < fl, but must reduce the loop gain neglecting performances in order to preserve stability at frequencies f > fl. Thus the bandwidth of the closed-loop system is defined by the intermediate frequency dynamics whose structure is known. Since adaptive controllers permanently estimate the parameter vector 0, they could be used to extend the bandwidth of the system to ]'2, thus extending the usefulness of the system model up to frequency f2. Such an adaptive controller must still provide enough gain attenuation at frequencies above f~ in order to preserve stability in the face of unstructured uncertainties. Obviously we need a frequency selective adaptive controller with the following properties: (i) achieving large loop gain at low frequencies (ii) adapting to unknown and possibly time-varying parameters in order to maintain stability and performances at intermediate frequencies ]'1 < f < f2, and (iii) providing attenuation to preserve stability at high frequencies, f >
f2. Such a frequency selective controller which will behave as an adaptive controller for some frequency range and as a nearly fixed controller in another frequency range will be proposed in the present section. Since transfer functions Wi(z),i = 1,2,3 are known, we are able to design the low-pass finite impulse response filter (FIR) l~(z), satisfying: 1) 1ITV(All2eil)l-~ 1 for f < f~, where A is a fixed number satisfying, At + A0 < A < 1, 0 < A1 < < A0, and A0 is specified by assumption ($1) 2) 1lTV(Al/2eil)l <__T'~o'for f > f2, where To = m a x 1_
{11 W,(=)liar*. }
(62)
or if we want good disturbance rejection at high frequencies To = 1max
and 7 is the upper bound on
{!!
}
IW~(At/~-eq)l,i -
(63) 1,2,3 at low frequencies
f~_f2. The underlying idea of this design is that at low frequencies f < ]'2,
W(All2eil) Bo(A1/2eil) Ao(All2eif)
A0(A1/2eil)
(84)
72
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
and for frequencies f > f2 i = 1,2,3
(65)
where 7 and ,~ are the same as in (61). For frequencies close to f~, conditions (64) and (65) may be difficult to achieve, since we need a filter with short transient band. In practice this is the matter of the designer skills and the specific system that has to be controlled. In our considerations we will assume that (64) and (65) are satisfied. We will also assume that W(z) is a stable polynomial. Specifically, if we design a filter with amplitude characteristics described by Eq. (64) and (65) and some of its zeros appear to be unstable, we will use a filter where the unstable zeros are replaced by their reciprocal values. These two filters have different phase characteristics but the same amplitude characteristics, and that is what we are interested in. Thus, the low-pass filter W(z) has the following form W(z -1) = w0 + wlz -1 + - " +
(66)
w, wz -"W
and we assume that (5'4) the zeros zj of t4Z(z) satisfy Izil _< )~1/2, where )~0 + )ix < )i < 1, and 0 < Ax < < ,~0 and ~0 is defined by assumption (Sx). We will stabilize the system (57) by designing the adaptive controller so that for a given reference signal y'(t), the following functional criterion is minimized J -
+ 1) - y ; ( t +
1)] +
2,
(sT)
where y/(t)-
ITV(q-1)y(t), y } ( t ) - I ~ ( q - t ) y ' ( t ) ,
(68)
while polynomials p(q-1) and Q ( q - t ) are chosen by the designer and are defined as follows
p(q-1)
=
l + ptq -1 + . . . + p,~pq-nP
Q(q-t)
=
qo + qtq - t + " ' ' + qnoq-n~
(69)
In (68), ~'V(q-1) is the low-pass FIR filter designed so that requirements (64) and (65) are satisfied and is given by Eq. (66). Using prior information related to the transfer function Be(z)/Ae(z) at low frequencies f < f2, we can choose polynomials p(q-1) and Q(q-1) so that the following assumption is satisfied:
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
73
(5'5) All zeros z1 of the polynomial
D(z) - Ae(z)Q(z) + So(z)bV(z)P(z) satisfy [zj[ __<)~1/2, where A0 + A1 < A < 1, and 0 < A1 < < A0, while A0 is specified by assumption ($1), and bowo + qo :/= O. We shall assume that reference signal y* (~) satisfies: ($6) {y*(t)} is a bounded deterministic sequence, i.e., there exits a number ml such that [y*(t)[ _< ml for t > 1. We assume that all initial conditions are finite. Note that the system model (57) can be written in the form
Ao(q-1)yf(t + 1) = Be(q-1)lTVu(t) + 7(t)
(70)
where "r(t)
=
[Be(q-1)A1(q-1)W1(q-1) + Ae(q-1)A2(q-l)w~.(q-l)]~V(q-1)u(t)
+
A3(q- 1) Wa(q-1 ) 1;k(q- 1)~(t)
(71)
It is not difficult to see that from (70) we can obtain e(t + 1)
where
-
[Bo(q-X)l:V(q -1) + Q(q-1)]u(t) - q[Ao(q-:) - 1]y1(t)
+ q[p(q-l)_ l]yl(t)_P(q-i)y;(t+ I)+ 7(t)
(72)
e(t + I) - P(q-1)[yl(t + I ) - y}(t + I)] + Q(q-l)u(t)
(73)
Equation (73) can be written in the form e(t + 1) = oTr
1]y/ (t) -- P(q-1)y~ (t + 1) + 7 ( t ) (74)
where
e0r -
;b0, h i , . . . , b..]
(75)
and
6(t) T = [yl(t),..., y l ( t - nA + 1); u l ( t ) , . . . , u l ( t - nB)]
(76)
u y ( t ) - ~V(q-1)u(t)
(77)
where From Eq. (74) it is obvious that when unmodelled dynamics and external disturbances are absent, control low optimal in the sense of (67) is given by 80Tr
+ Q(q-1)u(t ) + q[p(q-1) _ 1]y/(t) - P(q-1)y}(t + 1)
(78)
Based on the available prior information related to the system model at low frequencies f < f2, we introduce the following assumption:
74
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
(ST) The compact convex set O ~ which contains the true parameters 60, the sign of r0 = bowo + qo and a lower bound r0,min, on the magnitude of r0 are known. Without loss of generality we assume that r0 > 0 and r0,min > 0. For the estimation of 00 we propose the following algorithm ~(t + 1) - 7~{8(t)+ n,(t)r
+ 1)}, 0 < v < 2
(79)
where e(t) is given by Eq. (73), while 79{.} projects orthogonally onto (3~ so that :P{0} E O ~ for all/~ E ~,,~+-B+I, and there exists a finite constant do so that [[ 0(t) - a0 [[2_< do < oo and ~0(t) > r0,min > 0 for all t > 0. The algorithm gain sequence n4,(t ) is given by
n,(,) = ,2o + where the vector r satisfying
1)+ II r is given by Eq.
2, 1
< no < o o
(80)
(76), and A is the fixed number
A1 + Ao < A < 1
(81)
for 0 < A1 < < 1 and A0 is defined by the assumption (S1). The role of the number no in Eq. (80) is to prevent division by zero in algorithm (79), and is chosen by the designer. Since 00 is unknown, as the adaptive control law, we will use the "certainty equivalence version" of (78), i.e.,
#(t)To(t) + Q(q-1)u(t ) + q[p(q-1) _ 1]yl(t) = p(q-1)y~( t + 1)
(82)
It is not difficult to see that the estimation scheme, actually represents the algorithm for the identification of the parameters 0o of the nominal system model. Specifically, from Eq. (73) and (82) we conclude that e(t) in the algorithm (79) is given by e(t + 1) = YY (t + 1) - 8(t) r r
(83)
and represents the prediction error, which is usually used in all parameter identification schemes. In the algorithm (79) we are using measurement vector r and e(t) with filtered input and output signals. The purpose of this filtering is to attenuate the influence of the large, high frequency unmodelled dynamics. Since at high frequencies f > f2, unmodelled dynamics are unstructured and complex, they can not be estimated. Our goal is only to estimate the low frequency (f < f2) system dynamics. The obtained
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
75
estimates for the low frequency system dynamics do not have to be significantly affected by the high frequency unmodelled dynamics. This implies that at high frequencies, updating of the parameter estimates should be suspended (in the ideal case) or significantly slowed down. In other words, we need a frequency selective adaptive controller. In the subsequent analysis it will be shown that the proposed adaptive algorithm provides such properties. Estimation scheme (79) actually represents the normalized gradient algorithm with projection, where we use nr instead of c+ II r c > 0, as is usual in similar situations. The motivation for selecting such nr is to make possible the usefulness of the Robust Ultimate Boundedness Theorem, formulated in the previous section. Specifically, for the construction of the corresponding Lyapunov functions in the proof of the Theorem 3.1, we need property (Aa)(i) introduced in Section 2, and n~(t) given by Eq. (80) satisfies assumption (A3)(i). Let us define the following constants:
7z -
l[ Se(z)Wz(z)I?d(z) Jilt.., 72 -11Ae(z)W2(z)I~(z)[[aH ~ Ae(z) Bo(z)W(z) II Z~(z) II~, C D B - I I D(z) [!~
CD~
--
CDp
-- II P(=) II~|
,
CD~-Ii
Q(=) II~--
c~ap
-
II Ae(z)P(z)~V(~) Be(~)~(~)2P(_-) D(z) [1~**, CDBp =[I D(z) [[~/**
Cw
-
II w.(~)~r
(84)
wh~r~ II" I[~o. is the "compressed" H ~ norm defined by Eq. (5), X is given by (81) and D(z) is defined by assumption (5'5). Since assumptions (Sz) and (5'5) hold, all the above defined norms exist. Regarding the intensity of unmodelled dynamics we introduce the following assumption: (S8)
("/1
"~" ~[2 ) C D p <: 1.
The above assumption actually implies two requirements: 1) at low frequencies f <_ f2, Wi(z),i = 1,2, should satisfy condition IWi(A-Z/2eil)l < 7, where 7 > 0 is a small number, and 2) .at high frequencies f > f2, performances of the designed low-pass filter W(z)should provide [Wi(A-Zl2eil)]~r(A-X/~eil)[ < 7, i = 1,2. The following lemma will be useful for future reference. Lemma_ 4.1" Let the assumptions (Sx) - (Sa), (Ss), ($6), (Ss) hold. Then 1) (BeWP + AeQ)u(t) - Aee(t + 1) + AePy~(t + 1 ) - PT(t) (85)
2) (Bor
+ AoQ)yj(t + Z) = Ber
+ Z) + Bor
+ 1)+ Q~(t) (S6)
76
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
where the unit delay operator q-1 is omitted in all operator, while 7(t) and e(t) are defined by Eqs. (71) and (73) respectively.
3) II 7(t)I1~_< c~ II ~(t)IIx +k, + ~11($) (87') where II 7(t)Iix and II e(t)I1~ are defined by Eq. (1), when x(t) is equal to 7(t) and e(t + 1), respectively, while
(7, + 72)CDA
C~ = 1 -
(88)
(71 q" 72)CDP
and k~= (
1 [(7t -b 72)CDAPm1 -[" Cw 1 - (71 + 72)CDp
kw]
} ( 1 - 1A)t/2
4) II ~(z)I1~< c~ II ~(t)I1~ +k~ + e,2(t) where !! ~(t)!1~ i~ defined by Eq. (1), when z ( t ) - u(t), while C,, - CDA + CDpC.,,k,, = (1 - mlA)I/2 CDAP -[- ]r
(89) (90)
(91)
where C~ and k7 are defined by Eqs. (88) and (89), respectively. (92) 5) II yj(t)I1~< c~ II ~(t)Ii~ +~ + ~3(t) wh~,~ il y~(t)I1~ is given by Eq. (1), when ,(t) = y(t + 1), whil~, Cy = Cz, B + C~oC-,
(93)
and ml
CDBP q" kTCD Q.
(94)
In relations (87)- (92), constants 71,72, CDA, CDp, Cw, CDAP, CDB, CDQ and DDBp axe defined by Eq. (84) and the constants k~ and ml are defined by assumptions ($2) and ($6), respectively. Proof: The proof of the lemma is given in the Appendix. Important properties of the adaptive algorithm (79)-(82) will be formulated in the following lemma. L e m m a 4.2: Let the assumptions ( $ 1 ) - (Ss) hold. Then 1) ne(t) < C, n e ( t - 1)+ k, + ~14(t)
(95)
where n~(t)is defined by Eq. (2), when z(t) = e(t + 1) and e(~) is given by Eq. (73). The constants Ce and ke do not depend on the gain # of the algorithm (79), and will be specified in the proof of the lemma. At the same time, Co and ke depend on 7i, i - 1, 2, in such a way that their values decrease as 7i, i - 1, 2 decrease, where 7i i = 1, 2 are defined by Eq. (84).
DISCRETE-TIMEROBUSTADAPTIVECONTROLSYSTEMS 2) he(t) < Cr
77
(96)
C~2 or ~15($), 0 <: C~bl, C~2 <:
where the constants Cr and Cr do not depend on the algorithm gain/~. Proof: The proof of the lemma is given in the Appendix. We are now in a position to formulate the main (global stability) result. T h e o r e m 4.1" Let the assumptions ( S 1 ) - (Ss) hold, together with ($9) Cv < 1 - 2e, where C~ is defined by Eq. (88). Then 1) n~(t) < F1 where
(97) F1 =
1
F ~ ~2Cr + ~0
1 - ~2
C~I
(98)
where 0 < e2 < 1 and Cr i = 1, 2 are the constants from the Eq. (96) and are given by Eqs. (A.30) and (A.31) in the Appendix. Constant F is given by Eq. (15), where C~ and k.r are defined by Eqs. (88) and (89), respectively. The constants Co and ke in Eq. (15) are specified by relations (95) and (A.28) in the Appendix. The constant ~0 is the same as in Eq. (14). 2)
n,.,(t) < ny(t) <
2[C~F1 + k~] 2 2F,..2w(C~F1 or k~)
(99) (100)
where the constants G,,k~, C v and ky are given by Eqs. (91), (93) and (94), respectively and the constant F~ is defined by Eq. (98). Proof: From Eq. (79) it is not difficult to obtain
V(t + 1) < l / ( t ) + 2/~ ....nr
nr
--F- 1) 2
where V(~) --II 0(t)!! 2, and 0(t) = 0 ( t ) - 00
(101)
(102)
From (101) and Eq. (A.20) in the Appendix, it follows that -
~(t)
~
le(t +
1)!. 17(t)l
n4,(t)
(103)
i.e., we obtain the same recursive relation as (6). The global stability result simply follows from Theorem 3.1, if we prove that assumptions ($1) - ($9) imply assumptions (A1)-(A6), which are necessary for the Robust Ultimate Boundedness Theorem (Theorem 3.1).
78
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
Since V(t) < do < oo, which is ensured by the projection in the algorithm (79), we conclude that assumption (Ax) holds. Assumption (A2) follows from statement 1) of Lemma 4.2. Assumption (A3)(i) is automatically satisfied by the definition of n#(t) in Eq. (80). Assumption (Aa)(ii) is proven as the statement 2) of Lemma 4.2. Assumption (A4) simply follows from statement 3) of Lemma 4.1. Assumption (A6) is identical to assumption ($9). Let us now prove that assumption (As) introduced in Section 3, holds. Note that in assumption (As), S(I) is defined by Eq. (13), where variables e(t) and 7(t) are defined by Eqs. (73) and (71), respectively. Since from assumption (As), on the subsequence {ak},k = 1,2,3,..., it is true that
S(o'~ + 1) - ~(1 - ~- + C~)e(o'k + 1) ~ - 2~[e(o'k + 1)1" I"r(ok)l + .xs(,,k) > 0 (104) and S(ak) < O, we conclude that ( 1 - ~" + C.y) e(ak + 1) - 2l~(o-k + 1)l. I"r(,,k)l > 0
(105)
wherefrom it follows that
I"r(o'k)l _< 5I ( I _ P 7 + C,)-le(o'k + 11l.
(106)
Since from Eq. (A.20) in the Appendix,
I~(o-k + 1)1 _< d~12 II r
+l'r(,,k)l
(107)
by (106) we derive
:(1 + ~ - - C~)le(o'k + 1)[ < d~12 II r
(108)
Using the fact that S(ak) < O, from Eq. (104), we obtain
S(o'k + 1) < ~(1 - ~- -t- C.v)e(crk -'1"-1) 2
(109)
wherefrom by (108) it follows that
/.,(1 - ~ + c.,)
S(o'k -'1'-1) < 4do( 1 + ~ _ C..),)2
II r
II
(110)
From the previous relation we obtain that assumption (As) holds, i.e.,
S(o'k -I- i)
,,~,(,,'k)
(111)
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
79
where /~(1 - 2a + C~) Cs - 4d0 (i-I- a2- C.r);"
(112)
Thus we conclude that assumptions ( $ 1 ) - (5'9) imply assumptions ( A 1 ) (A6) introduced in Section III, and consequently by applying the Robust Ultimate Boundedness Theorem, statement 1) of the Theorem 4.1, follows directly. Substituting (97) into (90), relation (99) follows directly. Taking into account assumption ($4), relation (100) can be easily obtained from (92) and (97). Thus the theorem is proved. O A simple physical explanation of the mathematically described phenomena in Theorems 3.1 and 4.1 is as follows: unmodelled dynamics and disturbances can cause the estimated parameters to migrate ("drift") from the stabilizing set to the instability set, and as a consequence, ne(t) becomes very large. As a result of that instability, the bursting function S(t) defined by Eq. (13) becomes positive and the Lyapunov function V(t) + S(~)/nr 1) starts to decrease, producing ne(t) < Fx < c~. In other words, a large prediction error e(t + 1) gives to the estimator relevant information for correct identification of the system parameters which reenter into the stability set. With the parameters in the stability set, prediction error e(t + 1) decay, becomes less and less exciting. The drift phase may thus restart. In fact, it is the instabilities that are responsible for the global stability of the adaptive system. This phenomenon is known as self-stabilization. R e m a r k 1: (Performance of the algorithm when there is no unmodelled dynamics and external disturbances.) In the ease when 7(t) = 0, from (103) it follows that
(
/~ 21~ 1 - -~
e(t + 1 n~(t)
-< do < co
(113)
wherefrom by statement 2) of the Lemma 4.2, it is not difficult to obtain lim e(t + 1 ) - 0,
(114)
~---*OO
i.e., we achieve the global minimum of the functional criterion (67). R e m a r k 2: (Continuity property of the prediction error e(t + 1), with respect to unmodelled dynamics and external disturbances.) From (103), we can obtain
2u(1-
e(~ + 1 t--1
<eo+2.
"" le(t + 1)l-I;(t)l t=l
From the previous relation, two cases follow:
;,:;it)
(11.5)
80
MILOJE
S. R A D E N K O V I C
AND ANTHONY
N. M I C H E L
N
( p ) ~ e(,n,(t) + 1)2 1) p 1-- : 2)
( 1 -):
~~
~(t + 1) ~
n,(t)
,=:
(116)
< 4 ,=1 N n4,(t)'r(/)2
(117)
The first case is trivial, i.e., similarly as in the case of Remark 1, we can obtain limt_oo e(t + 1) - 0. Let us consider the second case. Since n4,(t) > no, from (117), we obtain
Ee(t+l) 2<-
1-7
t=l
"-
nO
max he(r)
l
t=l
7(t)2.
(118)
From Eq. (71) we can derive lim sup 1 A" ~,-oo ~, ~ ~ ~(t)~ -< ( ~ ~ + ~w~)~
(119)
t=l
where
7w
-
maxl[Bo(z)A~(z)W~(z) + Ao(z)A2(z)W~(z)]#(z)l, Izl=l
and
(120) (121)
Izl=z while cu and a~ are the upper bounds for h /
(122)
N---oolim~ ,..., u(t)2 <_ cu < oo t=l
and lim sup 1 ~
*
(123)
t----1
Substituting (119)into (118), we obtain lim sup 1 N lV--oo N N E e(t 4- 1)2 t=l
< "-
4K~ ~)2 (Twcu 4-cw(7w) 2 no(1-
where K~ denotes the upper bound given by max n~(r) _< K# < oo. l
(124)
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
81
From Eq. (98), it is not difficult to see that constant F1 can only decrease, when 7i, i - 1, 2 decreases. The same holds for the upper bounds of nu(t) and ny(t), given by Eqs. (99) and (100), respectively. Thus we conclude that constants K~ and c~ in inequalities (122) and (125) can only decrease when unmodelled dynamics and external disturbances disappear. Finally, when 7w "* 0 and c w ~ 0, i.e., when unmodelled dynamics and external disturbances go to zero, it is obvious from (124) that the mean-square prediction error goes to zero, thereby giving us robust performance. R e m a r k 3: (A small algorithm gain ~u may result in practically unacceptable, very large input and output signals.) In this remark we shall consider how the upper bounds for the functional criterion (prediction error) e(t), input and output signals, established in Theorem 4.1, depend on the algorithm gain ~u. Note that the integer L in Eq. (15) is given by Eq. (35), i.e., L-
/~(1 - ~ - C,)r
+ 1
(126)
where 0 < E1 < 1 is independent of p, and we used the fact that px is defined in assumption (As) in Section III. Substituting Eq. (112) into (126), we obtain
L(#)
do
4#
#u
C~) ~
+ I. (127)
Substituting (127) into (15), we obtain the dependence on the algorithm gain p, of the upper bound of the functional criterion (prediction error) e(t + 1) 2. From Eq. (90) we conclude that constant k.~ is independent of p. From Eqs. (A.30) and (A.31) in the Appendix, we see that the same holds for the constants Cr and C ~ . Constant 0 < e.~ < 1 is defined by Eq. (43), and we conclude that it is independent of #. Similarly, statement 1) of Lemma 4.2 establishes that the constants Ce and ]co are independent of ]~. Finally, from Eqs. (15) and (127) it is now obvious that there exists an optimal value/z0 for which the upper bound established in (14) (or (97)) is minimal. From Eqs. (15) and (127) we can derive lim F(/J) = +oo
(128)
lim F1 (/~) - +oo
(129)
D--0
and ~---0
where F1 is defined by Eq. (98). Therefore, from (97), (99) and (100) it is obvious that a small algorithm gain ~ may give rise to unbounded growth of the input and output signals. This fact can intuitively be concluded,
82
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
without previous analytical analysis. Simply, let us consider the behavior of the algorithm in the time intervals [uk, rk+l), where the bursting function S(t + 1) is positive (see Eq. (31)). These time intervals correspond to the fast adaptation intervals of the adaptive system, which is explained in the next remark. Thus in the fast adaptation intervals, from Eq. (31) we derive Eq. (34), and this implies that the algorithm stabilizes itself at almost all points from the the time interval [uk, rk+l), except at the finite number of points defined by Eq. (35). It is obvious that if ~a is very small, the number of the unstable points L, is very large. Consequently, a small p implies slow adaptation and a large number of unstable points. In other words, in the fast adaptation intervals [~k, rt+x), a very small p does not allow the adaptive system to use the advantages of the large and exciting prediction error signal e(t + 1), thus making self-stabilization more difficult. In the extreme case when p --~ 0, the number of the unstable points L(p) ~ +co, without a chance for self-stabilization. In the previous analysis we assumed that the intensity of unmodelled dynamics C~ is fixed and satisfies assumption (5'9). From the same assumption it is also obvious that the smaller the p is, the larger the unmodelled dynamics C~ that can be tolerated. R e m a r k 4: (The proposed adaptive algorithm acts as a frequency selective controller.) At the beginning of adaptation the parameter estimates are incorrect, generating large prediction errors which are larger than 7(t) defined by Eq. (71). In such cases, the bursting function S(t + 1) given by Eq. (13) becomes positive. Generally, whenever high frequency unmodelled dynamics and external disturbances are excited, the estimator (79) produces incorrect parameter estimates. As time progresses, these estimates escape from the stability set and the prediction error e(t + 1) starts to grow unboundedly. As a result of that instability, the bursting function S(t + 1) becomes positive. Those periods of system operation where S(t + 1) > 0, in our analysis correspond to the time intervals [~k, rk+l). From Eq. (29) it is obvious that then we will have fast adaptation. This is because as pointed out in Remark 3, the algorithm gain p does not have to be chosen small. As a consequence of the fast adaptation, the prediction error rapidly decreases and approaches zero. Since e(t + 1) - - 0 and unmodelled dynamics and external disturbances are always present, though small, they can cause S(t + 1) (Eq. (13)) to become nonpositive. When S(t + 1) _< 0, the Lyapunov function V(t) =11 00 If2 may start slowly to diverge and the algorithm (79)-(82), will generate incorrect parameter estimates. Such estimates can lead adaptive systems slowly into unstable regimes and excite high frequency unmodelled dynamics. The signals generated in the closed-loop system under such conditions do not provide reliable information for correct updating of the parameter estimates in the algorithm (79).
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
83
Therefore, the Lyapunov function V(t) continues to diverge. These periods of adaptation correspond to the time-intervals [rk, at) in the analysis presented in Section III. Ideally, in those time intervals, updating of the parameter estimates have to be suspended, and the controller have to be with fixed parameter, obtained at the end of the adaptation process in the time-intervals [ak-1, rt). Next we will show that in the time-intervals [vk, irk), proposed adaptive controller behaves as a "nearly" fixed controller with slowly changing parameters, under the assumption that 71,72 and Cw given by Eq. (84) are small. From Eq. (79) we obtain
II #(t + 1)- #(t)I1=< u= + 1) 2 ,,,(t)
(130)
wherefrom we derive t
Z At-j l[ d~(j + 1) - 0(j) j=l
112< --
(131) no
where he(t) is given by Eq. (8). Since in the intervals Irk, trt) the bursting function S(t + 1) > 0, we conclude that (27) is valid for all t E [r~,tr~). Thus from (131) and (27) we obtain for t E [rk, ak) t
4p~k~ E A' - j [I O(j + 11 - O(j) [[2< no'(1 ' a2 --'C.y) 2 + ~16(t)
(132)
j-1
where C~ and k.r are defined by Eqs. (88) and (89) respectively. Recall that the designed filter l~(z) provides at high frequencies f > f2 attenuation 7/T0, where To is given by Eq. (63) and 7 is the intensity of the unmodelled dynamics at low frequencies f < ]'2 (Eq. (61)). Our assumption that there exists a transfer function Bo(z)/Ae(z), which well matches the system dynamics at low frequencies implies that 7 is small and consequently the constants 71,72, C~ and C~ are small. Suppose that we choose no = 1 and so that p + 2C~ < 1. Since C~ is small, such a choice of/~, still does not imply that p is small, i.e., p < < 1. It follows from Eqs. (132) and (89) that the controller parameter changes are small in the time intervals [rt, ak), i.e., the adaptive controller behaves as the nearly fixed controller. Note that relation (132) tells us that in the intervals [rt, try), the parameter estimate can slowly drift and escape from the stability set, thus generating large prediction errors e(t + 1). Then the bursting function S(t + 1) becomes positive and fast adaptation (self-stabilization phase) again takes place. Fast adaptation over the time intervals [rk, ak) is the undesirable property of the adaptive system. This can be explained as follows. Over the time intervals [at, rt+l) where the bursting function S(t + 1) > 0 we have
84
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
fast adaptation and the estimator generates correct parameter estimates. If the algorithms possesses fast adaptation over the interval [vk, ~rk), correct parameter estimates obtained in the interval [at-1, rk) can be significantly deteriorated. Specifically, over the time intervals [rk, crk), the bursting function S(t + 1) < 0 and from the analysis presented in Section III, it is obvious that then the Lyapunov function V(t) may diverge. We now propose two methods of enhancing the slow adaptation over the time intervals Irk, ak). The first method requires prior information related to the constants "11,72, Cw, CDA, COp and CDAp defined by Eq. (84), and the upper bound k~ of the noise w(t). In the next remark it is pointed out that based on the available prior information related to the physical system, the constants 7x, 72, CDA and Cop can be roughly estimated by the designer. The same holds for the constants C~, Cw and ChAp. Thus, if we know k~, we can estimate k~ defined by Eq. (89) and design the following estimation algorithm: over the intervals where ne(t) > 4 k ~ / ( 1 - ~ - Cv) 2 we are using small no in Eq. (80) and in the intervals where n,(t) < 4k.~/(1- a _ Cv)2 we can use estimation scheme (79) with no > > 4~u2k~/(1 - ~ - Cx)2, or completely stop adaptation. The second method for obtaining slow adaptation over the intervals [rk, cry) is to modify the sequence n#(t) in the algorithm (89). Specifically, instead of nr given by Eq. (80), we now define
he(t) = An~(t-- 1)+ II r
=
2
+
1-A
(133)
and we obviously require the upper bound kw of the noise w(t) to be known. Such choice of the sequence n,~(t) does not influence the results derived in Sections III and IV, except that constants Co, k0, C~2 from Lemma 4.2, are different. Since in the intervals [rk, ~rk), the bursting function S(t + 1) < 0, from Eq. (13), we obtain
e(t + 1) 2 _<
(1
,
4
n.r(t )
+
(134)
wherefrom by (130) we conclude for t E [rk, ~rk) t
j=~
t
[[ 0(J + 1) - 0(j) il 2<
- i l - ~ + c~)-~ .=
~'(J)
he(j)"
(135)
From Eq. (71) we derive _<
+
+ Cw.
(t +
2+
(136)
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
85
where the constants 71,72 and Cw are defined by Eq. (84). By (134) and (135) it is not difficult to obtain for t E [rk, ~k) t
Z
II 0(J + 1) - 0(j) I]< 70(t - rk) + ~ls(t)
(137)
j --'1"k
where
4# 2 70 - (1 - ~ + C.v)2 [71 + 72 + Cw] 2.
(138)
It is not difficult to see that 70 is small and that slow adaptation takes place in the time intervals [rk,ak). Thus, incorporating nonfiltered input signal u(t) in the sequence n~(t) can be useful. The final conclusion is that the proposed adaptive controller possesses to some extent frequency selectiveness property and if we have sufficient prior information related to the real physical system, such property can be enhanced, resulting in a more robust adaptive system. R e m a r k 5: Note that assumption (5'9)implies assumption (Ss) and actual restriction of the admissible unmodelled dynamics is the assumption ($9). This assumption requires 71 and 72 to be small enough. Using the fact that the low-pass filter W(z) does not influence the unmodelled dynamics at low frequencies and attenuate it at high frequencies, it is obvious that the constants 71 and 72 will be small if the unmodelled dynamics is small enough at low frequencies. This implies that transfer function Be(z)/Ae(z) should match very well the system dynamics at low frequencies, which is required kby the assumption ($3). Since the transfer functions Wi(z), i = 1, 2, 3 are known, by using prior information related to the transfer function Be(z)/As(z) at low frequencies, the constants 71,72, CDA and CDp can be estimated by the designer. This means that the assumption (5"9) can be roughly checked before implementing the adaptive algorithm on a real physical system. Finally all conditions for the global stability are given in the frequency domain, which is more acceptible for engineering practice.
VQ
ROBUST ADAPTIVE OUT BURSTINGS
CONTROL
WITH-
Generally, the fact that an algorithm is globally stable does not imply that the direct practical application of the considered algorithm is possible. From the analysis presented in the previous sections it is obvious that the bursting phenomena are possible during all of the system operation time. In the periods of bursting, the signals in the closed-loop system can become unacceptably large for satisfactory operation of the physical system. Therefore, there is a need for the design of the robust adaptive control without
86
MILOJE S. RADENKOVIC AND A N T H O N Y N. M I C H E L
the possibility of burstings to occur. It seems natural that for such a design we need much more prior information about the physical system than the set of prior information under which global stability is proved. For the adaptive control presented in Section IV, we do not need to know the intensity of the unmodelled dynamics C.~, except in the case when the designer wants to check validity of the assumption ($9). In order to design adaptive control without bursting we have to know the constants C 7 and k-y given by Eqs. (88) and (89), respectively. From Eq. (89)it is clear that for estimating kT, the prior information related to the H ~176 norms 71,7~, COP,CDAP,Cw and upper bound kw of the noise sequence w(t), are required. As pointed out in Remarks 4 and 5, using the available prior information about the physical system, some time it is possible to estimate the constants 71,'72,CDA,CDP,CDAP and Cw. Specifically, all of these constants depend either on the known "high frequency" transfer function Wi(z), i = 1, 2,3 and transfer functions chosen by the designer, such as P(z), Q(z) and W(z), or on the low frequency system dynamics Be(z)/Ao(z) for which we always have some prior information. Thus, if in addition we know the upper bound of the noise kw, constants C-r and k.y can be estimated by the designer and in the present section we assume that they are known. In Remark 4 we noted that the reason for bursting is the drift of the parameter estimates in the time intervals Irk, ak). Ideally, in those intervals an adaptive controller should behave as the fixed controller. We will now show that knowing the constants C.r and k.r, makes possible the design of the adaptive control algorithm which guarantees that asymptotically bursting will not occur. Let us consider the following estimation scheme
ur e(t + 1) m,(t)
+ a)-
(139)
where me(t) is given by
m (t) -- mo + ,%(t), ,% =
X)+ il r
2,
0
< ,no <
oc,
(140)
where A < A,,~ < 1 and A is the number given by Eq. (81). From Eqs. (139) and (A.20) it is not difficult to see that
V(t+l)
-
2 / J ( 1 - ~ u ) e ( t + l ) 2 ~ 7(t) ~
m (t)
m,(t)
(141)
where V(t) is defined by Eq. (102). At the same time, from Eqs~ (A.1) and (A.2), we obtain 7(t) 2 _< 27"(t) 2 + ~a6(t) (142) where t
7" (t) 2 = Z At-j [C.~e(j + 1) 2 + k~x] j=l
(143)
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
and
87
k~, - ( 1 - A)k~
(144,}
where k.r is defined by Eq. (89). We consider the case kto ~ 0 and consequently kpr ~ 0. Substituting (142) into (141) we obtain V(t + 1) <_ V(t) - 2/~(1 - ,u)
,(t +
1)' +
27"(t) = + {36(t)
.
(145)
Following the same line of reasoning as in Section III, it is not difficult to see that for the recursive relation (145) we can define the following bursting function So(t + 1), for t >_ 1, t
So(t + 1 ) - 2 Z A ~ - i j=l
{~{1-~u)-
2 ] e ( j + 1) 2 - 7*(j) 2}
(146)
where p0 is an arbitrarily small number, i.e., 0 < ~ < ~u(1- p) and is chosen by the designer. Since we assume that the constants C v and k.r are known, 7" (t) and consequently So(t + 1) can be measured by the designer. When the bursting function So(t + 1) is positive, then the recursive scheme (145) converges. In the intervals where S0(t + 1) < 0, drift of the parameter estimates is possible, and the updating of the parameter estimates should be suspended. This fact suggests the following adaptive algorithm for t > 1
~(t + 1) = ~(t) + m~(t) p.6(f) e(t + 1),
if So(t + 1 ) > 0
/~(t + 1) =/~(t),
if So(t + 1) _< 0
(147)
where 0(I) - 01, IIOx II_
(Sx0) p2 = p(1 - / ~ ) -
1
po
C$
;~
2 > 0
A.
The following robust stability result can now be established. T h e o r e m 5.1: Let the assumptions ( S x ) - (S'6), and ($10) hold. Then
1) where
lim m~(t) <
t--cc
-- P2(1
)I)(1
,,
)im)
(148)
t
,n,(t)- Z k~-Je(J + 1)2
(149)
j-1 while kl-r is given by Eq. (144) and p2 is defined by assumption ($10), 2) there exists the finite integer To, so that/~(t + 1) = 0(T0),
fort>T0
88
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
3)
lim
(z50)
me(t) = O.
t --- c o
k~ ---, 0 71,72 "-* 0 P r o o f : Two cases are possible1) s0(~ + 1) < 0 for all t > 1 2) So(t + 1) changes its sign during the algorithm operation. In the first case the statements of the theorem follow trivially from the definition of So(t + 1) (Eq. (146)) and Eq. (147). This actually means that the initial value 0(1) is a good estimate of the real parameters. Before proceeding further with the analysis, let us show that the case S0(t + 1) > 0 for all t > 1 is impossible. From (145) we obtain ^
V(t -I- I)+ So(t -I- I) < V(t) + So(t) Ie(t )____+___~~ ~36(t) m#(~) m , ( t - I) -P~ m#(t) ~ m#(t) (151) wherefrom it follows that lim e(t + 1 ) 2 / m # ( ~ ) - O. Similarly, as in Lemma 4.2 from Eqs. assumption ($7)
(152)
(85) and (86), we can obtain by
,n,(t) _
< oo
(153)
where me(t) is given by Eq. (149). From (152) and (153) we derive lim,-oo e(t + 1) = 0. Then from Eqs. {:143) and (146) we conclude that there exists 0 < T1 < co, so that for t > T1, So(t + 1) < 0 by which it is proved that the case S0(t + 1) > 0 for all t _ 1 is impossible. Let us now consider the case when So(t + 1) changes its sign. Similarly, as in the proof of Theorem 3.1, we define the sequences rk and ak as follows -- T1 <0"I < T2 < " - ' < Tk < O'k < Tk+l < ' ' "
(154)
so that S0(t + 1) > 0, S0(t + 1) _< 0,
for t E [r
rt+i)
for t e [rk,ak),k -- 1 , 2 , 3 , - . - .
(155)
If ,5'0(2) > 0, we set n = 0 and al = 1. From (146) it is not difficult to see that p~(1 -- A)(1 -- Am)
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
for t E [r~, ~r~). After summation from t = ~r~ + 1 to t from (151) that V(7"k+l) -[-
_<
= Tkq.1 -"
89
1, it follows
~0 (7"k'l"1 ) mr 1)
(157)
r~~-I
V(ak + 1 ) + S0(ak + 1)
e(t + 1) 2
,=o~+1
rk+a-x ,=ok+l m~(t)
Note that from Eq. (146)
So(ak + 1) = 2Lu(1 - / ~ ) -
2]e(ak + 1) 2 - 27"(~rk) 2 + AmSo(o'~:) (158)
and from (145) e(erk -i- 1) 2 V(~rk -I- 1) _< V(rk) -- 2ju(1 --/~) m~(crk) + 27"(~k) § 536(crk)
(159)
where we have used the fact that 0(ak) - 0(rk), which follows f r o m algorithm (147) and the definitions of the corresponding intervals in (155). Since by definition (155), S0(ak) < 0, after substituting (158) and (159) into (157), we obtain 'e(t + I) 2
k•--
So(~+,) < v(~k)- po ,=-k V(Tk+l)-I- 1Tl~b(7.k+ 1 - I) -
rk+x --I
m,(t) + ,=ak ~
~3~(t) -
-
-
-
-
m~(t)
(160)
Note that from (155) it follows that S 0 ( T k + I ) > 0. Then from (160) we derive oo -k+,-1 e(t + 1) 2 P0 Z Z m~(t) < V ( 1 ) + c~ < oo (161) k=l t=a~ for TI = i, or oo
ra,+x-1
k=2
,=o~
)2 m,(t)
< v ( ~ 2 ) + ~ < oo
(162)
for vl - 0 and al - 1 and the constant c~ is the consequence of the initial conditions. The proof that V(r2) < c~ is simple and is based on the argument that in the interval [crl, r2),S0(t-t- 1) > 0. Then from (157) we obtain for k = 1, and crk = 1,
v(n) < v(2)~ s0(2) -
m~(1) F cr < oo
(163)
90
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
where 0 < ce < oo allows for the capture of the effects of the initial conditions. Relations (161) and (164) imply " k ~ ' - 1 e(t "~- 1)2
2i-.m =,.
aT(~
(164)
=~
wherefrom it follows that I r,,+, 1 lim Z e(t+l) ~-0 k-oo too(k) ,=o~ -
(165)
where
too(k)-
max
m~(r).
(166)
From (153) we conclude that r~+1-1
too(k) < C§ Z
e(t +1) 2 + Cr
(167)
- 1 ) + Cr
t~-Ok
and from (156) and (165) we can obtain rh+x - 1
lira
k---oo
Z
e(t+l) 2 - 0 .
(168)
t-'qk
This implies that there exists K0 < c~, so that for k > K0 t
Z
Atm-ie(J+ 1) 2 < k~.rl2
(169)
j--~O'k
for t E [uk, r~+l). Note that from Eqs. (143) and (146) we can obtain
So(, + 1)_< Z
j~ok
+
+
(170)
for t 6 [uk,rt+1). Using the fact that So(ok) <_ O, we conclude from relations (169) and (170) that S0(t + 1) _< 0 for t E [crk,rk+1) and b _> K0. Therefore, So(t + 1) _< 0 for all t _> rKo- W e have thus proved that in the estimation algorithm (147), updating of the parameter estimates will be suspended after time t _> 7Ko. By this statement 2), the theorem is proved. Since for t >_ rKo, S0(t + 1) _< 0, statement 1) of the theorem follows simply
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
91
from Eq. (146). Statement 3) of the theorem is the trivial consequence of statement 1). Thus the theorem is proved. !:3 Theorem 5.1 establishes that the upper bound of the performance index is bounded and that the parameter estimates are constant after some finite time To. Thus for t > To drift of the controller parameters is eliminated, i.e., we have a fixed controller which implies that bursting can not occur for t > To. These performances are achieved without ~r-modification or projection in the estimation algorithm.
VI.
CONCLUSION
Using the concept of the bursting function, new insights into the robust adaptive control problem are obtained. For the estimation of the controller parameters, a normalized gradient algorithm with projection is proposed. Global stability of the adaptive system is established for a class of unmodelled dynamics and external disturbances which are unstructured, complex and large at high frequencies. It is shown that the small algorithm gains/~ may result in unacceptably large input and output signals. The presented results also demonstrate one possible way of incorporating a pr/or/system information into the robust adaptive control design. For the sake of simplicity, the system unit delay case is considered in this chapter. It is not difficult to see that using the proposed methodology, similar results can be established for the general delay" case.
VII.
APPENDIX
P r o o f of t h e L e m m a 4.1- The first two statements of the lemma follow by simple manipulations of Eqs. (70) and (73). Let us prove the rest of the lemma. From the definition of 7(t) (Eq. (71)), by assumption (S~), it is not difficult to see that
II 7(t) IIx< ('r, + 7.~) II u(t) II~,+ c ~ II ~(t) I1~,+69(t)
(A.1)
where [I z(t)I1~, z q. {7, u,w} is given by Eq. (1) in Section II. Constants 7i, i = 1, 2 and Cw are defined by Eq. (84). Similarly, from Eq. (86), we can obtain
II ~(t)i1~< eva [Ie(t)I1~ +CoAp [[u'(t)I[~ +cop II 7(0 I1~ +6o(0 (A.2) where II y'(t)IIx is defined by Eq. (1) when z(t) = y'(t + 1). The H r162 norms CDA, CDP and CDAP are defined by Eq. (84). Statement 3) of the
92
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
lemma can easily be obtained by substituting (A.2) into (A.1), where we used the fact that assumption (5"8) holds and II ,,,(t)IIx_<
ko~l(1
-
A) 1/2, II y'(t)IIx_< roll(1 -- A) 1/2.
(A.3)
Statement 4) of the lemma follows by substituting (A.1) for (A.2). Let us prove statement 5) of the lemma. From Eq. (85), we can obtain
II y1(t)IIx~ cz~B II e(t)IIx +Cz~Bp II y'(z)IIx +CDQ II "r(t)I1~ -']"~21(t) (A.4) where the constants CDB, CoO and CDBP are defined by Eq. (84). Substituting (A.1) for (A.4), statement 5) of the lemma follows directly. Thus the lemma is proved, o P r o o f of L e m m a 3.2: From Eq. (82) it follows that I(l)o(t) wo + qo)ul(t)) <- d'o
lul(t - 01 + ~
lYI(t + 1 - i)l
i=1
+~
1 + ~ Ip, I ml
IqiT-lu(t - i)i + ~ Ip, i-lyf(t + 1 - i)l +
i--1
]
i-'l
(a.5)
i=1
where d~ is the upper bound for the 0(~), i.e., II 0(r d~ and it is provided by the projection scheme in the estimation algorithm (79). Since from Eqs. (66) and (77)it follows that
luI(t)i < ~ Iw~l. lu(t- i)1
(h.6)
i=0
we obtain from (A.5)
nu,(t) < glnu(t
-
1) + g~ny,(t) +
K3
"1-~22(t)
(A.7)
where n~(t), z E {u I, u, y! } is defined by Eq. (2) in Section II, and
wfA
nW
+
.,
(A.s)
q~A -i
r0,min i = 0
K~
=
4 n Ad~
A-i +
--
4
"-W'---r0,min
"P
}
O,min i=1 2
i=1 /~3
np
1 q-
IPi] i= 1
m~ 1-A
(A.IO)
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
93
where dl - d ~ / r 0 , m i n , while d~ is given as in (h.5), and r0,min is the lower bound for the i)o(t)wo + qo. The existence of these bounds is ensured by the projection mechanism in the algorithm (79). Since from Eq. (80), 9, y ,
. , ~ , (t) + , o +
~.~3(t)
(A.11)
by using relation (h.7) and (h.ll), we derive
he(t) <_K4ny,(t) + Ksnu(t - I) + K6 + ~24(t)
(A.12)
where
K4 -
~.= A-i +
~-i K2
Ks u ( "i~=oA-') lil
(A.13) (A. 14)
rSB
K6
-
K 3 ~ A -i + n 0
(A.15)
i--O
where the constants Kx, K2, and K3 are defined by Eqs. (A.S), (A.9) and (A.10), respectively. Using the fact that the low pass FIR filter l~(z) is stable, from (A.7) we obtain u
F
F3
(A.~6)
where
Kw =11 W(')-i II~--
(A.17)
From relations (A.1) and (A.3) we can obtain
n.y(t) < 2(71 + 72)2nu(t) + 2C~ where n.y(t)is defined by Eq. (2) when into (A.18) we derive
k~
i - A + ~2s(t)
x(t) -
(A.18)
7(t). Substituting (A.16)
n.,(t) _< 2(71+ 7~.)2KiK~vnu(t- I)+ 2(7a +7~.)2K~K~n~j(t) + (71 + 7~):KwKs + 2Cry i - A+ ~2v(t)
(h.19)
94
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
where the constants Ki, i = 1,2,3 are defined by Eqs. (A.8) - (h.10). Since Eqs. (72)and (82) yield
e(t + 1) - -O(t) Tr
(A.20)
+ 7(t)
where 8 ( t ) - 0 ( ~ ) - 8o, we can obtain
he(t) < 2don,(t) + 2n,(t) + ~28(t)
(A.21)
where n.(t) is given by Eq. (8) and do is the upper bound for II (t)-0o do < c~ provided by the projection in the algorithm (79). Substituting (A.12) and (A.19) into (A.21) we can obtain
n.(t) < K, nv,(t) + Ksn.(t - 1) + K9 + ~29(t) where
K~ = Ks -
2[doK4+ 2(71 + 72)~KwK2] 2[doKs+ 2(71 +72)2KwKI]
(A.23) (A.24)
and K9 - 2d0Ks + 4[(71 + 72)2K~, + C~, 1
A]"
(A.25)
Note that from relations (90) and (92)
n.(t) ___2c2n.(,) + and
+
n~,(t) < 2C~n,(t- I)+ 2k~ + ~sl(t).
(A.26)
(A.27)
By relations (A.26) and (A.27) from (A.22) it is not difficult to obtain
n,(t) <_2(C~K7 ~ + C~Ks)ne(t- I) +2(k~/'KT+ kuK8 ) 2 + K9 + ~a2(t) (A.28) where the constants Ki, i = 7, 8,9 are defined by Eqs. (A.23) - (A.25). The constants Cu,k,,,Cy and kv are given by Eqs. (91), (93) and (94), respectively. It is obvious that by relation (A.28), statement 1) of the lemma is proved. From the previous proof, it is evident that the crucial fact for establishing the "exponentially growing rule", is the boundedness of the parameter estimates, which is ensured by the projection mechanism in the algorithm (79). It is not difficult to see that all constants which appear in (A.28) are independent of the algorithm gain p. At the same time all constants in (A.28) are either independent of 7i, i - 1,2 or when they depend on
DISCRETE-TIME ROBUST ADAPTIVE CONTROL SYSTEMS
95
7i, i = 1, 2, then as 7i, i = 1, 2, decrease so do the constants. Let us prove the second statement of the lemma. From Eq. (77) we can obtain
n,,(t) < nw (~=o w~A-i) nu(t) + ~33(t)
(A.29)
where nw = deg I~r and the wi are the coefficients of the FIR filter W(z). Substituting (A.29) into (A.11) we obtain
,~,(t)+
A
nw
(t)+no+~z4(t)
(A.30) By substituting (A.26) and (A.27) into (A.30) it is not difficult to derive
nr
<_C~1 ne(t) + Cr + ~as(t)
(A.31)
+ c2,,w
(A.32)
where c,, = 2 i=1
\i=0
and
C,2=2
k.2v
A-i + k~,nw ,
A-i
w~A -i
+ no
(A.33)
i=1
Thus the lemma is proved.
References [1] P. A. Ioannou and J. Sun, "Theory and Design of Robust Direct and Indirect Adaptive-Control Schemes", Int. J. Control, 47, pp. 775-813, (1988).
[2] B. Egardt, Stability of Adaptive Controllers, Springer-Verlag, New York, N.Y. (1979). [3] B. E. Ydstie, "Transient Performance and Robustness of Direct Adaptive Control", IEEE Trans. Aulomai. Control 37, pp. 1091-1105 [4] S. M. Naik, P. It. Kumar and B. E. Ydstie, "Robust Continuous Time Adaptive Control by Parameter Projection", IEEE Trans. Aurora. Conlro137, pp. 182-197 (1992).
96
MILOJE S. RADENKOVIC AND ANTHONY N. MICHEL
[5] M. Radenkovic and A. Michel, "Robust Adaptive Systems and Self Stabilization", IEEE Trans. Automat. Control 37, pp. 1355-1369 (1992). [6] M. Radenkovic and B. Ydstie, "Using Persistent Excitation with Fixed Energy to Stabilize Adaptive Controllers and Obtain Hard Bounds for the Parameter Estimation Error", to appear in July issue of SIAM J. Contr. and Optimization (1995). [7] M. Radenkovie and J. Bialasiewiez, "Robustness of Unmodified Stochastic Adaptive Control Algorithms", IEEE Trans. Automat. Control, 39, pp. 396-401 (1994).
A Two-Riccati, Feasible Algorithm for Guaranteeing Output L Constraints Guoming G. Zhu Cummins Engine Company, Inc. Columbus, IN 47202-3005
Robert E. Skelton Space Systems Control Lab., Purdue University West Lafayette, IN 47907
I. I N T R O D U C T I O N In the presence of all L2 disturbances whose outer products are bounded by a known matrix N, this chapter develops controllers to minimize the worst case control L norm bound subject to the worst case L constraints on each output group. The resulting controller is an LQG controller with a special choice of the output weighting matrix and input noise intensity matrix. Both weighting matrices may be determined by an iterative algorithm. The design algorithm turns out to be feasible if the algorithm converges. It has been shown in [ 1-3,10] the standard Output Covariance Constraint
(OCC) problem has two different interpretations; stochastic and deterministic. One deterministic interpretation of the OCC problem is the output L constrained control problem ( O L ) , described as follows. Consider the system CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
97
98
G U O M I N G G. Z H U A N D R O B E R T E. S K E L T O N
A Px P + B Pu + D
P
where
xp e 9~"~,
yp
=
Cpx p
Z
=
M xp
y p ~
9~"~,
z~
P
wP
(1.1) +v,
p
9~~ ,
ve
9~": ,
u~
9~"" , and
wp ei 9~"
are,
respectively, vectors of the plant state, plant output, measurement signal, measurement noise, control input and deterministic exogenous signal. We assume that the closed loop system with some static or dynamic controller u(s) =[C (sl - A)-' B c + D]z(s)
is strictly proper, then it can be expressed in
the following state space form .~
=
Ax+Dw
; w e g l "~"~
y
=
Cx ; ye91""',
(1.2)
w h e r e C = [ C r , C [ ] r , w = [ w p, r v r] r and y = [yp,u r r ] r , and matrices Cy a n d C . depend on the coordinate of the closed loop system state. Here we assume that the plant performance outputs are divided into m output groups y, = C x ~ 91" (i = 1,2, .. . , m ) , that is yp = [ y , ~ ,Y2~ ..... Y.~]~ . Let the square of the L 2 norm of w(.) be def'med by
IIw<
(1.3)
(t)W-'w(t)d,,
=
where W is a symmetric positive definite matrix, and define the L
norm of
each output group y, (-) to be
Ily,<>11
yr (t)y, (,), i : 1,2 ..... m,
(1.4)
and note that
II.,<.>ll = up.
, (t),
i=
1,2 .....
nu.
(1.5)
Define the L 2 disturbance set (1.6)
GUARANTEEING OUTPUT L CONSTRAINTS
99
Then the O L problem can be stated as follows. THE OL PROBLEM .o
For the given plant (1.1)find a (static or dynamic) controller to minimize
s ug.llu,r
JoL = E r i=1
n:}
; r > 0 , i = 1,2,. .. ,nu
(1.7)
subject to
Ily, )ll <
i=1,2
,m, V w es
w
(1.8)
where r > 0 (i = 1,2 ..... nu) and e, > 0 (i = 1,2 ..... m) are given constants.
E! The significance of this problem is obvious: under the given L2 disturbance environment we seek a controller which satisfies the output L constraints with minimum control effort J o,.. This is a practical controller design problem. In physical hardware there are always hard limits due to sensor and actuator signal saturation, and such systems (e.g., the Hubble Space Telescope) must be evaluated based upon its ability to keep pointing errors less than values specified by optical resolutions. Such missions might not be feasible if such L inequality constraints cannot be guaranteed. The O L problem has been solved in [3,10]. Also, the O L design algorithm has been applied to the "NASA Minimast' flexible structure lab facility at Langley Research Center and the results have been reported in [ 1]. Often, in practical problems, much more information about the disturbance vector w(.) is known than simply the L: norm in (1.6). For example, we may know bounds for the L2 norm of each w, (-). To improve the O L method, we presume to have more information about the disturbance than the L2 norm bound in (1.6). Suppose we know that w(.) belongs to the following disturbance set s
= w(-): t ~ 9~''+'z ,
w~ (t)dt < Ix, 9
One can still use the OL results over the disturbance set .a
(1.9)
100
GUOMING G. ZHU AND ROBERT E. SKELTON
= {w(.):,
IIw()ll:
=
w" ( , ) W - ' w ( t ) d t
< ~},
(1.10)
nw+nz
by choosing 13= ~ Ix~ and W = I, that is, to enlarge the disturbance set K2 so i=l
that ff~ _c ~ . It can be easily shown that the set (1.10) is always equal to or large than the set (1.9). Therefore, the old O L approach may yield more conservative designs than the present paper. The outer product of all the input disturbances, which is a positive semidefinite matrix, will include this additional information (1.9). Hence, we presume to know a bound N>O on the outer product of w characterized by the following set f2N= {w(-): t --->9~--',
(
foW(t)w r (t)dt< N}.
(1.11)
The precise statement of the new problem, called the Extended Output L EOL ) constraint problem, will be given in the next section. The chapter is organized as follows. Section II provides the definition of and some preliminary knowledge which will be useful for
the EOI_, problem
solving the problem. Section III provides the continuous time solutions (necessary conditions and an algorithm) for the static controller (measurement feedback) and dynamic controller. Two special cases, state feedback and strictly proper full order dynamic solutions are discussed in Section III. Section IV is the discrete time version of Section III. Section V gives an example for a discrete system to demonstrate the algorithm and to compare with the O L designs. Section 6 adds some conclusions.
II.
STATEMENT OF THE
Before precisely stating the Lemmas. Lemma 2.1
EOL EOL
PROBLEM problem, we introduce two very useful
GUARANTEEING OUTPUT L, CONSTRAINTS
101
For the asymptotically stable and controllable system (1.2) define Y = CXC r, where X is the positive definite solution of the following Lyapunov
Equation 0 = XA r +AX +DWD r, W > 0 .
(2.1)
Let w(.) in (1.2) be any L 2 disturbance defined in (1.3) and let
Ily(.)ll = s u p y t~>O
T
(t)y(t)
(2.2)
be the square of the L norm of y(.). Then
Ily(-)ll ---
IIw(-)ll:,
(2.3)
and for all disturbances belonging to the set ~w in (1.6)
13. trj = suplly(-)ll . wr163w
(2.4)
Lemma 2.1 was proven in [4,9].
Remark 2.2 The bound given in (2.3) is for the asymptotically stable and strictly proper system (1.2). Note that for the proper (not strictly proper) system, i.e., there is a direct feed forward term in y(t), hence, there exists no L bound on the output for any disturbance which belongs to the set f2 w in (1.6) with ~ > O. Therefore, the closed loop system with finite L norm on the output for any w(.) ~f2 w should be in the form (1.2). Lemma 2.3 Let (1.3) and (2.2) respectively denote the square of the input L 2 norm and the square of output L norm of system (2.1). Under the same assumptions as in Lemma 2.1, consider the disturbance w(.) in the set (1.11), where N>O is ^ given. Suppose there exists a W such that
102
GUOMING G. ZHU AND ROBERT E. SKELTON
Y
YYo
=
L(L~L)-V2 L; N = LL;
=
CXC ~; 0 = X A r + A X + D W D r,.
=
Dr ~oea~,C r yoYoCe r ~,d t D > O ;
(2.5)
[Ylyo; Ilyoll=1.
:
Using W for W in (1.3), (2.3) leads to "~[Y]tracel~-' = sTIly(.)ll'.
(2.6)
A proof is provided in [6]. Remark 2.4 Equation (2.6) shows that the bound (2.3) will be tight over the disturbance set O N by the choice o f W as the L 2 norm weight in (1.3) to
replace W. In the E O L problem we consider disturbances in the set (1.11). It is easy to see that for any L 2 disturbance w(.) which belongs to ~N defined in (1.11), we have
=
~oo wr (t)W-~w(t)dt trace{W-"2!w(t)wr(t)dtW-"2} (2.7)
<
trace (W -''2NW-''2)
=
traceNW-t,
where W > 0 is a given weight matrix. Therefore,
Ilu,<)ll ]y,
<
[ C XC,r ],, traceNW-I
<
"o[C XC r ]. traceNW -~ , i = 1,2,...,m,
,
i = 1,2 ..... nu, (2.8)
where X satisfies (2.1) and C = [ C r , C r ..... c rT] . and C, are defined in (1.2). Then the E O L
EOL Problem:
problem can be stated as follows:
GUARANTEEING OUTPUT L CONSTRAINTS
103
Find a static or dynamic controller and the input L 2 norm weighting matrix W > 0 to minimize the EOL cost J eo~. = traceNW-' . { traceRC XC r }
(2.9)
subject to (2.1) and "o[C~XCr].traceNW -' < e 2 i = 1 , 2 , i
'
~
,m,
(2.10)
where e, (i = 1,2 ..... m) are given and R =diag[rz,r 2..... r ] > 0 .
(2.11) E!
The physical interpretation for the E O L problem is clear if one considers the inequalities in (2.8). Note that in this case
Er,
(2.12)
Ilu,
i=l
Hence, the cost function is an upper bound of the weighted summation of supremums of control signals over all L2 disturbances which belong to ~N" The EOL| problem is similar to the O L problem, but the input disturbances belong to different sets, f2 N and fir, respectively. The next section provides the solutions of the E O L problem in the continuous time case.
III. S O L U T I O N S
OF CONTINUOUS
TIME EOL
PROBLEM
In this section, we shall discuss the first order necessary conditions for static and dynamic controllers, and also present an iterative algorithm. A.
STATIC MEASUREMENT FEEDBACK CASE
For the static measurement feedback case, we seek a constant feedback gain matrix G which is the solution of the E O L problem such that
104
GUOMING G. ZHU AND ROBERTE. SKELTON (3.1)
u =Gz.
Therefore the EOL| problem becomes (3.2)
I min traceNW -~. trace R G M X M r G r P
G.W>O
P
subject to (2.1) and
[o[C,XC,r ].traceNW-I - -
<
2
9
_ e,,
t = 1,2 ..... m.
To keep the closed loop system strictly proper, we only consider the case where v = 0 for system (1.1). In this case, the closed loop system (1.2) matrices are
A = A +BpGMp, D = D Theorem
, W =Wp, C=[Cp,Mr].p /.
3.1
Suppose that G is an optimal solution o f the EOL EOL
(3.3)
problem
is
regular.
Then
there
exists
problem, and that the a
matrix
Q = block
diag[ Q~ , Q2 ..... Q. ] > 0 such that i)
G = - ' f f - ' B r K X M r ( M , X M pr)-,.,
ii)
0 = KA + A r K + M r G r ' R - t G M p + CrQCp;
iii)
DrKD
iv)
Q{block diag[ C, x c r , C2 x c j ..... C x c r ]. t r a c e N W -' - F} = 0;
v)
A X + XA r + D pWD pr = 0 ,
P
P
= W - ' N W - ' ( t r a c e R G M X M rGr + traceQCpXC r)''2" P
P
(3.4a)
where "R = R. traceNW-' , Q = Q. traceNW-' , F = diag[ 2, I,,~.212 2 ..... 1~I 2 ], and l j is an identity matrix with dimension mj.
The proof of the above theorem is obvious. By using the following augmented cost function
H
trace NW -' { trace RGM pXM r G r } + +trace K[ X ( Ap
i=! + BpGMp ) r + ( Ap +
[ C x c ,rtrace NW -' - E I , (3.5)
BpGM, ) X
+ OpWD p r ],
GUARANTEEING OUTPUT L CONSTRAINTS
105
setting the partial derivatives with respect to X, G, K, W to zero, and adding one equation for inequality constraints in (3.2) (Kuhn-Tucker conditions), one can get the necessary conditions. By setting M = I , one obtains the results for state feedback case which is a special case of measurement feedback control.
Corollary 2.2 Suppose that G is an optimal solution of the EOL problem for the state feedback case, and that the EOL| problem is regular, that is, the EOL problem is not independent of the EOL cost function in (3.2). Then there exists a block diagonal positive semidefinite matrix Q such that i)
G = -'R-'BrKX;
ii)
0 = KA,, + A r K - KB,-R-'B rK + Cr'QCp ;
iii)
D ~, rKD
iv)
Q{block diag[ C, x c r , C~XC[ ..... C x c r ]. traceNW -' - F} = 0;
v)
(Ap + BpG) X + X (Ap + BpG) r + DpWDp r
P
m
p = W-I NW-I (traceRGXG r + traceQC X C ; ) |'~"
(3.6)
"-0,
g
where R , Q and F are the same as those in Theorem 3.1. The solution of the third equation in (3.3) and (3.6) is given in the following Lemma.
Lemma 3.3 Suppose that N and K are positive definite and D has full column rank. Then equation D r KD = W-INW-1 has one symmetric positive definite solution W~ = K~'[K~NK~] ~2K~-',
(3.7)
and one negative definite solution
w, where
r; ',
(3.8)
106
G U O M I N G G. Z H U A N D ROBERT E. S K E L T O N
(3.9)
K,~ = ( D r K D ) ''2 .
The proof was presented in [6]. An algorithm used to solve those necessary conditions is presented in Subsection C. B.
DYNAMIC FEEDBACK CASE Consider the following dynamic controller with order nc xc
=
AcXc + Bcz
u
=
C x +Hz.
(3.10)
It is well-known that by setting
=[APo ;x=
(3.1 la) =
;~=
;G= o
s
Ix:] [':] [5] ;y=
~
;D=
;D
=
x
"
;I)= B,,
; A
[01
(3.1 lb)
'
with compatible dimensions, the closed loop system can be written as (3.12) y
----
[d' , ,~'c" l" X-F H v
In order to obtain a bounded output L
9
norm for an L2 input, system
(3.12) should be strictly proper, i.e.,/4 = 0 or v - 0. Here we consider the case that v = 0. Then the closed loop system is in the form (1.2) with A=A+BGtr
C
[~r
21~/rGr]r
(3.13)
Let (3.14)
GUARANTEEING OUTPUT L CONSTRAINTS
107
where R = diag[r~,r2..... r ] > 0. By solving the following static measurement feedback EOL problem
trace NW -l . trace [CGl~I X~Ir G r
min
I
G.W>O
(3.15)
P
subject to (2.1) and
---
"
^T
[o[CXC ].traceNW
-1
~
2
_ e,, i = 1,2 ..... m.
one can obtain first order necessary conditions for the dynamic controller case. Hence, when v = 0 in (1.1), the EOL problem with a fixed order dynamic controller produces the static measurement feedback EOL controller as a special case. Special Case: Full Order Dynamic Controller For this special case, we consider the following strictly proper full order dynamic controller L
=
ax+
u
=
Cx.
z
(3.16)
Let
X =
IXPx-~Xc]; Y = [ ~ I ; w=IWvl'
(3.17)
then the closed loop system can be put into the form (1.2) with A -BM
I
BM c
D=
The EOL follows:
A +BC
A +BM
p
P 0
-BM
c
C;C= n
c
P 0
-A
(3.18a)
p
P.
(3.18b)
c
problem with full order dynamic controller can be stated as
108
GUOMING G. ZHU AND ROBERT E. SKELTON
min
traceNW-'.traceRC XC r, C = [ 0 , C ]
Ac ~c .Cr .W>O
(3.19)
subject to (2.1) and
"0[ C XC r ]. traceNW-1 <_ e 2,, i = 1,2,... ,m. where, in this case, C = [ C, C p ], C, = [C, r , C r ..... C r ]r.
Theorem 3.4 Define the following matrices
w
=
r
;K=
[ ll
;X=
[xl x2] I
(3.20)
,
where Wj, K o and X o (i, j=l.2) are nx by nx matrices. Suppose that the triple ( C , A c, B ) is the optimal solution to the full order dynamic EOL problem defined in (3.19), and that the EOL| problem is regular. Then there exists a matrix Q = block diag[Q~ , Q2 ..... Q. ] > 0 such that Q{block diag[ C, x c r , C2 XC r ..... C XC r ]. traceNW-' - F} = 0,
(3.21)
and A =A,+B
C-B
M,;
(3.22)
B = ( X t , M pr + D W~2)W2-2" "~-i
C=-R
T
B K22,
where X , , and K22 satisfy X,, A ,r + A , X ,, _ ( X I M,
,r +D Wn)W~-l (X,,
M ,r
+D W~2)r +D W~,D ,r =~,
(3.23)
K=Ap + ArK= - K=B "R-'Br K= + Cr'QC = 0 , and "R = R . traceNW-' and Q = Q . traceNW-l . In addition DrKD = W-'NW-' (traceRC XC r + t r a c e Q C XCry )~2, where K is defined in (3.20) with K~2 = K22 and K~2 satisfying
(3.24)
GUARANTEEING OUTPUT L CONSTRAINTS
109
(3.25)
r ( K , , - K2:) + C~rR'C~ =0,
(K,,-K22)(A,,-BM)+(A,-BM)
and X is defined in (3.20) with X,2 = 0 and X22 satisfying
(3.26)
X22(A +B C ) r + ( A p + B p C ) X 2 2 +BW~2Br =0.
The proof is similar to the static measurement feedback case. Consider the augmented cost function m n
_~.
(3.27)
traceNW-' . traceRC XC r,, + E ' ~ [ C XC r 9traceNW-' - E,] I, : i=l
+traceK .[ AX + XA r + DWD r ].
Then (3.27) can be rewritten into H = traceXCrO.C + traceK .[ AX + XA rDWD r ] _ traceQ. F,
where 0 = block diag['Q,'ff ] and "R = R . traceNW-' , "Q = Q . traceNW-' . By setting the partial derivatives with respect to K, X, W to zero and adding the Kuhn-Tucker condition (3.21),one can complete the proof. _...
.__
Note that (3.22) and (3.23) describe an LQG controller if Q, R, and W are given. Hence, the full order E O L controller is an LQG controller with some special choice of the weighting matrices Q, R and input noise intensity matrix W. (The forthcoming algorithm determines Q, R and W). Remark 3.5 The fact that there exists a W > 0 satisfying the first order necessary conditions requires that D rKD > O. Remark 3.6 The first order necessary conditions for the OL problem only need to satisfy (3.22) and (3.23) for fixed W, which are included in the EOL| problem. Hence the OL| problem is a special case of the EOL problem.
C.
NUMERICAL ALC~RITHM
110
GUOMINGG. ZHU AND ROBERTE. SKELTON
Subsections A and B of this section provide all the necessary conditions for the design of static and dynamic controllers. The following algorithm provides a way to design the controller by iterating on these necessary conditions. This is an extension of the O L algorithm presented in [3].
THE EOL .o ALGORITHM Step I
Given system matrices Ap,Bp,Cp,Dp. Mp, initial weighting matrices Wo , Qo , R , outer product bound N, output L bounds e,, (i = 1,2 ..... m), error tolerance e , and free parameters 0 < [~< 1 and a >0. Define ~ = R. traceNW, -! ~ = Q. trace NW -~ Let i = 0
Step 2
Controller design step: Case 1- static state feedback Compute m-!
G, = - R
7"
(3.28a)
BpK,
by solving (3.28b)
K i A p + A ; K - K B p R---' i Bp"K -t- C;'QiC p = O, and then solve for X, by (Ap+BpG,)X, + X,(Ap+BpG,) r +DpWD r =0.
(3.28c)
Let D, = Op, q = Cp, and C~ = G,. Go to Step 3. Case 2: Static measurement feedback Iterate on the following three necessary conditions to obtain the optimal measurement feedback gain G i for given Qi and W
6,
- R -, ' B prK , Xi M pr ( M P X, M rP ) -''' O= K(Ap+BpG, Mp) r + K ( A p + B p G , Mp)
(3.29a)
=
+ M;<
C,M
(3.29b) ,
0 = (Ap + BpG,Mp) X, + X, (Ap + BpG,Mp )r +DW~D r
(3.29c)
GUARANTEEING OUTPUT L CONSTRAINTS
111
Let D i = Dp, Cy = Cp, and C~ = G i Mp. Go to Step 3. Case 3: full order dynamic controller
Solve for controller matrices A~, B~, and C~ by A= i Ap+
BpC~ - B M i p, " M ,r + DpWs
8:=(x:,
(3.30a) '
;
C' = - g -i' Bp"K~2, '
and X,; At+, ApX[, + DpW/~Drp _ _ ( X I li T
i
(3.30b)
M rp+ DpW/~)(W2~2)-'(X[,M rp+ D W~;)r = 0; i
i
---1
T
i
K=A + A K= - K~:B R, B K= + C ~ , C
= O.
Then solve for K:, - K~2 and X~: by (K:,-K~2)(A P - B ; M P ) + ( A P - B ~ M P ) r ( K : , - r ~ )
(3'30c)
+<~,c =o,
i
x=(a p
+BC;/~+(A+Bp C'IX= +Ew~(8;/T =o, c i
i
i
where
Di=
[o P
;K,=
)r
;X=
ix: o] l
(3.30d)
i "
Let C = [Cp, C ], and C~ = [0, C'c ]- Go to Step 3. Step 3
Compute K; = (D, TK,D, )1/2 ;
I.2, = traceRC: X, (C: )r + traceQ~C X,C r ,
(3.31)
112
GUOMING G. ZHU AND ROBERTE. SKELTON
Then solve for W§
(3.32)
)-' (K~' NK~ )"' (K~)-' ~+ (1-~)W,
Q , ( i + l ) = e , -~y,,, (i)Q,(i) ya. ,,Y=C,
XC, r , j = 1 , 2 ..... m,
a,+, = block aiagIQ, (i), Q: ( i) ..... Q. (i)]. Step 4
/:
and
IIw,.,-w,ll< ,
,,op,
where
e:l~ F = block diag[e,l,, " ~ : ..... e : l ] . E l s e , i = i + l a n d g o t o S t e p 2 . D During each iteration of the special cases, state feedback or full order dynamic controller, the resulting controller is an LQ or LQG controller. Hence, in those two special cases, under the usual stabilizability, detectability assumptions the stability of the closed loop system is guaranteed during each iteration. Remark 3.7 The algorithm is written for 3 different cases. One makes a choice in Step 2 for different controller complexity. Lemma 3.8 Suppose that the EOL
algorithm converges to some weight matrix
W > O. Then the resulting controller is feasible, i.e., the closed loop system with the resulting controller will satisfy the inequality constraints (2.10) The proof of Lemma 3.8 is similar to that in [3].
IV. DISCRETE TIME VERSION Here, we consider the following discrete system
xp(k+l)
=
A x(k)+Bu(k)+Dpwp(k)
y,(k)
=
C x,(k)
z(k)
=
M x (k)+ v(k),
(4.1)
GUARANTEEING
OUTPUT L
CONSTRAINTS
113
where the dimensions of the matrices and vectors are analogous to the continuous system (1.1). Suppose that the closed loop system with a static or dynamic controller is strictly proper, then the closed loop system can be expressed as follows
x(k+l)
=
Ax(k)+Dw(k);we91
y(k)
=
Cx(k) ; y ~ R ~+~',
n"*n~
(4.2)
where C=[C~,Cr] r , w = [ w r , v r l r and y = [y~r,u~lr . Define the square of the input e: norm of w(.) as
IIw<-)ll: = ~ w, o
(4.3)
k=O
and let the square of the output e| norm for each individual channel be
Ily,<.)ll~.= s u p y r ( k ) y , ( k ) ; Ilu,<)11: = suo k - ~ u:t
i : 1,2 ..... m;
i = 1,2,
" ' "
(4.4)
,nu.
For the strictly proper closed loop system we have the following Lemma.
Lemma 4.1 [5] For the asymptotically stable system (1.2) with (A, D) controllable define Y = CXC r, where X is the positive definite solution of the following Lyapunov equation X = AXA r + DWD r, W > 0.
(4.5)
Let w(.) in (4.2) be any e: disturbance which belongs to
a~ -
w<.): k -~ ~t -+~,llw<.)ll: = E w~
and
Uy<-)II: - s~py~
(4.6)
114
GUOMING G. ZHU AND ROBERT E. SKELTON
be the square of the ~| normx~f y(.). Then
(4.7)
Ily(.)ll <_ trlllw(.)ll . and for all disturbances belonging to f~w in (4.6)
otY] = s~. ~y(.)~2| .~_,.
(4.8)
Define the input disturbance set f2 N ={w(.): k-'~91""~'~w(k)wr,~o ( k ) < N, N > 0 } .
(4.9)
The discrete time EOL| problem can be stated as follows. DISCRETE TIME EOL PROBLEM .a
Find a static or dynamic controller and an input matrix W >0 to minimize the EOL cost J EoL. = trace NW -~ "{ trace R C, x c r }
2
norm weighting
(4.10)
subject to (4.5) and "o[QXCr].traceNW -~ <_e 2,, i = 1,2,...,m,
(4.11)
where e, (i = 1,2 ..... m) are given and R =diag[r~,r 2..... r ] > 0 .
(4.12) t3
The same physical interpretation for the E O L problem as that for continuous time case can be found by using Lemma 4.1, i.e.,
(
i=1
and
wlit~ N
u:t
(4.13)
GUARANTEEING OUTPUT L CONSTRAINTS
]ly,<>11<-6[CXC,r].traceNW -~, i = 1,2 ..... m.
115
(4.14)
Hence, J EOL., is an upper bound of the weighted summation of supremums of the control signals over all the f 2 disturbances which belong to f~,, in (4.9), and inequality (4.14) guarantees that the output g norm for each individual channel will be less than or equal to the given bound. A.
STATIC MEASUREMENT FEEDBACK CASE
We seek a constant feedback gain matrix G such that the solution of the E O L problem is (4.15)
u =Gz.
Here we only consider the case that v = 0 to have a strictly proper closed loop system. After combining equation (4.1) and (4.15), the closed loop system (4.2) matrices are A = Ap + BpGMp, D = Dp , 9
and the discrete time E O L
r J'~IT
W = Wp , C = .t ~_ p , M r
(4.16)
problem for measurement feedback controller
becomes (4.17)
f min traceNW -~ 9traceRGM P X M PTG r G.W>O subject to (2.1) and o [ C XC, ]. traceNW -~ < _ E2, . . i. = . . 1. 2,
--
r
m.
Theorem 4.2 Suppose that G is an optimal solution o f the EOL problem defined in (4.17), and that the EOL problem is regular. Then, there exists a matrix Q = block diag[Q~, Q~ ..... Q= ] > 0 such that
116
GUOMING G. ZHU AND ROBERT E. SKELTON
i)
G - - ( R-- + BrpKBp) -' BrpKAFXMFr ( M pX M r ,) -l",
ii)
K = ArKA + MrG r ('R+ B;KBp)GMp +CrQCp;
iii)
D ; K D p = W - I N W -I ( t r a c e R G M p X M p r G r + trace __ QCp X C r )~,2.,
iv)
Q{ block diag[ C, X C r , C: X C r ..... C X C r ] . trace N W -' - F} : 0;
v)
A X A r + DpWD r = X ,
(4.18)
where - - = R . traceNW-', Q = Q. traceNW-', F = diag[e . 2I I ! ,e.212 2 ..... R
e21
],
(4.19)
a n d l j denotes an m j x mj identity matrix.
The proof of Theorem 4.2 is similar to that of Theorem 3.1. By setting M = I, one obtains the necessary conditions for the state feedback case, which is a special case of measurement feedback control. Corollary 4.3 Suppose that G is an optimal solution o f the E O L case, a n d that the E O L
f o r state f e e d b a c k
p r o b l e m is regular. Then there exists a matrix
Q = block diag[Q t , Q2 ..... Q. ] > 0 such that i)
G = -('R + BrKBF) -'BrKAp ;
ii)
0 = ArKAp - ArKBF ('ff+ BrKBp) -'BrKAF +Cr'QCp;
iii)
D pr K D p = W-' N W - ' ( t r a c e R G X G r + t r a c e Q C X C ; )
iv)
e { b l o c k d i a g [ C t X C r , C 2X C r ..... C X C r ]. t r a c e N W - ' - F} = 0;
v)
( Ap + B p G ) X ( A p + B pG ) r + D pW D rp= X ,
(4.20)
u2 .
where R , Q a n d I" are the s a m e as in (4.19).
The solution of the third equations in (4.18) and (4.20) can be solved by Lemma 3.3. The algorithm used to solve those necessary conditions will be given at the end of this section. B.
DYNAMIC FEEDBACK CASE We consider the fixed order dynamic controller
GUARANTEEING OUTPUT L CONSTRAINTS
x(k+l)
=
A x(k)+Bz(k)
u(k)
=
C x ( k ) + H z(k).
117
(4.21)
Let v = 0. Then the closed loop system has the following form x(k+l)
=
y(k)
=
[~r l~IrGr]rx(k)+[_iv(k),
(4.22)
where x(.), y(.), ,4, /~, /), C and M are defined in (3.11). The fixed order dynamic controller design yields the standard static measurement feedback problem, as a special case. Special Case: Full Order Dynamic Con~oller Consider the following strictly proper full order dynamic controller x(k+l)
=
A x(k)+Bz(k)
u(k)
=
C x (k).
(4.23)
By defining the closed loop system state vector x, output vector y and input vector w as (3.17), and defining the closed loop system matrix A, D and C as (3.18). The discrete time E O L problem with full order dynamic controller can be stated as follows f A~.B,.C~.W:,O min trace N W - ' . trace RC u XC ur; C,,= [0,C c] subject to (2.1) and
(4.24)
"o[C, XC 7 ]. trace NW -t < e.~, i = 1, 2 .... m. Theorem 4.4 Partition W, K and X as in (3.20). Suppose that the triple ( C , A c, B ) is an optimal solution o f the EOL problem defined in (4.24), and that the EOL problem is regular. Then there exists a matrix Q=blockdiag[Q~, Q2 ..... Q, ] > 0 such that Q{block diag[C~ XC r, C 2XC r ..... C XCr., ]. traceNW-' - F} = 0,
(4.25)
118
GUOMING G. ZHU AND ROBERT E. SKELTON
and A = Ap + B,,C - B Mp,
(4.26a)
) -'. B c = (ApX,,M pr +D,W~:)(W22 + MpX,, M T ,, C = -(R + BrK22Bp
-l
T
n K22A ,
where X,, and K~2 satisfy
(4.26b)
Xl, = ApX,,A r + DpWi,D r P
P
- ( A p X, , M rp +OpW~ 2)(W22 + g p x , ] Mr)-l(ap X , Mrt, +DpW~2) r r r K22 = ApK22A p - ApK22B,, (-~ + B r K22Bp )_, BpK22A r + C Pr QC, P P m
and R and Q are defined in (4.19). In addition, DTKD =W-'NW-' (traceRC XC[ +traceQC XC T ) II2 ,
(4.27)
where K is defined in (3.20) with K,2 = K22 and K n satisfying (K,, - K 2 2 ) = ( A p - B Mp) ~(K,, -K2:)(A p - B M p ) + C [ ' R C ,
(4.28)
and X is defined in (3.20) with X,2 = 0 and X22 satisfying X22 = ( A p + B p C ) X : : ( A p + B p C ) r +B (W:2 + MpX,,M~)B[.
(4.29)
The proof is similar to that of Theorem 3.4.
C. NUMERICAL ALGORITHM Step I
Given system matrices A, Bp ~ Cp ~ Dp.M p , initial weighting matrices Wo,Qo,R,
outer
product
bound
N,
output
L
bounds
e,, (i = 1,2 ..... ny), error tolerance e , and free parameters 0<13 < 1
and 1I >0, define R~ = R . trace lVW~-' ~ = Q trace NW. -~ Let i Step 2
Controller design step: Case 1: static state feedback
O.
GUARANTEEING OUTPUT L CONSTRAINTS
119
Compute a i -" --(R-~
"~
Brr, Bp) -' Brr, Ap
(4.30a)
by solving K, = ArK, A - A r K , B , ( ~ +BrK, Bp)-'BrK, A +Cr~Cp P
P
"
P
P
Pi
(4.30b)
P
and then solve for X, by (4.30c)
X, =(Ap+BpG,)X,(Ap+BpG,) r +DpWD r. Let D~ = Dp, Cy = Cp, and C'u = G,. Go to Step 3. Case 2: Static measurement feedback
Iterate on the following three necessary conditions to obtain the optimal measurement feedback gain G, for given Q~ and W G, =-(R~ + Brr, Bp) -' BrK, ApX, Mr (MpX, Mr)-';
(4.31a)
K, = (Ap + BpG, Mp) r K (Ap + BpG, Mp)
(4.31b)
+ MrG , ( ~ + BrKiBp)G, Mp +Cr~Cpp r X,=(Ap+BpG, Mp)X,(Ap +B pG,M p)r +D pWD , p.
; (4.31c)
Let D i = Dp, Cy = Cp, and C~u= G~M r. Go to Step 3. Case 3: Full order dynamic controller
Solve for controller matrices A i B ~ and C ~ by c ~
c p
c
(4.32a)
A: = Ap + BpC: - B: Mp ; , pr + DpW~;)(W:': + MpX,, M~) p -' 9 B: = ( ApXllM T
i
-1
T
i
C: = - ( R + BpK~2Bp) BpK22Ap, and
120
GUOMING G. ZHU AND ROBERT E. SKELTON
'
XI!
-- A p X 11 i A pr
-~-
D p W l ,' D pr
(4.32b)
X,,' M Pr + DpW/t)(W:'2 + M P X,,, M Pr)-I (ApX,,, M Pr + DpWl2, ) r . , s2 , B p ( ~ + B r K22B , r , p K n' A p - A p K p) -, BpK~sA +C ;~ C
- (A P
Kn'
-A"
P
i
P
i
"
)
i
Then solve for K,, - Ks2 and X22 by
(K[,- K~2 ) = (Ap - B~ Mp) r(K[,- K~s)(Ap - B: M) + C r ( ~ + B p KT s 2, B p ) C c
Xss' = (Ap + BpC;)Xs:(A i
(4.32c) )
p + BpC~ ) r + V~ (W~'2 + M p X , ,' M rp) ( B 'c) r ,
where
[o ,:]
D,= o
B~ ; r , = (K;,) ~ K;, ; x =
ix: o] o
(4.32d)
x~,
Let C = [Cp , Cp ] and C~ = [0, C~ ]. Go to Step 3. Step 3
Compute ,
D r
)us;
(4.33)
k', = traceRC~ X, (C~ )r + traceQ, C X,C r , Then solve for
w,., = x, (K;)-' (K; NK;)"' (K',)-'l~ + (~- 13)W,;
(4.34)
Qj(i + 1) = E-~Y j j~ (i)Qj(i) Y~" j , Y = Cj XC r, j = 1,2 , .. . ,m, Q,§ = block diag[Q, (i), Qs (i) ..... Q, (i)]. Step 4
,f
and 2
F =blockdiag[e,l,,
,op,
where
2
e212 ..... e S l ] . Else, i =i + 1 and go to Step 2. !"1
GUARANTEEING OUTPUT L CONSTRAINTS
121
Lemma 4.5
Suppose that the discrete time EOL algorithm converges to some weight matrix W>O. Then the resulting controller is feasible, i.e., the closed loop system (4.2) with the resulting controller will satisfy the inequality constraints (4.14). V.
EXAMPLE
In this section we will present an example to illustrate the EOI_, algorithm and to make comparisons with the O/., design. The example is a simply-supported Euler-Bernoulli beam of length L with a force and torque applied at the 0.45L point from left end. Using the beam model in chapter 3 of [8] and taking the first 5 modes of the model one can put the plant into the following state space model .;c
=
A x
yp
=
CpXp
z
=
M xp
P
P
+B
P
p
P
u+D
P
w
P
(5.1) +v,
where the output yp is the vertical displacement and rotation angle at the point of force and torque application which is at the 0.45L point from left end. The measurements are the vertical velocity and angular velocity at the same point (i.e., the sensors and actuators are collocated). Also, we assume that wp and v are L2 disturbances and wp is applied at the same position as that of actuators (hence, Dp = B )p . The design results for the continuous and discrete time controllers are similar, so we only present the discrete design results. For the discrete time controller designs, we choose the sampling time T = 0. 05 second and discretize the continuous model (5.1) to obtain the following discrete time model. x (k+l)
=
A xp(k)+Bu(k)+Dpw(k)
y,(k)
=
Cpxp(k)
z(k)
=
M , x (k)+v(k),
(5.2)
where system matrices Ap, Bp, Dp, C and Mp for discrete system (5.2) are presented in Appendix A. Note that the system matrices Ap, Bp, Dp, C and
122
GUOMING G. ZHU AND ROBERT E. SKELTON
Mp (5.2) are different from the system matrices in (5.1) .Under the assumption that the closed loop system input w(.) is constant during the sample period we have
IoW(t)w r (,)dt = s fj~i*"rw(t)w r (,)dt = T E w ( k ) w r (k). k~O
(5.3)
k=O
Equation (5.3) provides the connection between continuous and discrete outer products. A.
DISCRETE TIME STATE FEEDBACK CONTROLLER DESIGN
For the state feedback controller design, we assume that v = 0. Note that the dimension of wp is 2. Suppose that the components of wp satisfy
[[wp,(')~i = ~oWp' (t)wr~ (t)dt < 1~ [[wp2(')~i = Io wp2 (t)w;~ (t)dt < 1 ,
(5.4)
and
IoW ~(t)w;2 (t)dt = 0 . Using (5.4) we can have
ol
wp (k)w r (k) <
ol
=
1/T
= N.
(5.5)
20
Note that N is the upper bound matrix for the closed loop input disturbance w, and in the state feedback case w = wP . We choose, (5.6)
R =diag[1,1]; e = 10 -~', a = 1.5; [3= 1.0; F =diag[O.l,O.1]
as design parameters. The outer product bound on the input L2 disturbance is defined in (5.6). See Table 5.1 for results, Table 5.1 State Feedback Controller Design Controllers
"OL - 1 .a
0.10
[ 0.10
OL - 2
0.10
0.10
EOL
0.10
[ 0.10
GUARANTEEING OUTPUT L CONSTRAINTS
"-2
-'2
""2
"-2
U,
0.0999
0.2380
123
0.10
0.10
0.10
0.10
i.21x 103 1.72x 10~ 1.57 x 102 1.16x 102il.54x 102 1.17 xlO 2
U2 W
:4.55•
Q
I
~ 1
0
Jot,. or J cot.
%<s~ ~5,8))
N(see (5.5)) 82.9 0 1.27x109
0
1.73x10 s
"81.2
0
0
186
184
2.72x102
2.72x102
|
.
,,
17 Iteration # 10 16 The algorithm has not yet converged. With the choice W = I the best output g bounds that can be reached by state feedback are [0.0028, 0.2375]. Hence the algorithm would not converge since the solution does not exist. .
.
.
.
.
where
~',
m
2
(5.7)
= c, xc~, ., u, =[cxc~],,., i= 1,2,
1
Pole Plactm~at of EOL..inf and OL-inf
0.8 ...................... 0.6 .......... 0.4
~.2
~.6
..........
~.8 -I -I
......................
~.5
0
O~
l
Figure 1 Closed Loop Pole Positions for State Feedback
124
GUOMING G. ZHU AND ROBERT E. SKELTON
and the optimal weighting matrix
2.4436 -0.03721 Wsr = -0.0372
(5.8)
0.2513"
The resulting E O L controller guarantees that for any s 2 disturbance satisfying (5.5), the square of the output t ' norm of each output will be bounded by 0.1, and Table 5.1 shows that the E O L designs keep the outputs within their bounds using smaller control efforts (lower design cost E O L ) than the O L designs. If one chooses W = N in the O L design, the optimal cost of the O L design is close to that of the E O L design, and Figure 1 shows that the closed loop poles of the two design cases are very close. On the basis of our examples, W = N is a good choice for W in O L designs. To appreciate the disadvantage (conservativeness) in the O/., design, a choice W = 1 for the O L design yields unachievable design goals, (in this case when MF = 1 the lower bounds
on
the
e,,2 (i = 1,2)
achievable
in
(4.17)
is
[CDWDrCr],, p p p p
traceNW-', 9 but if one changes W to N for the O L design, the same design goals are achievable.
4000
C o n v e r g e n t Process of q I & q2
Solid line: q l D a s h e d line: q2 3000 ................ i.................. .................................... I
9 I
9
. 2 0 0 0 ....... i ......',~. 9
.
......................
.,.,,,,w
I
1
~
'
~
.....................................
0
:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Fig. 2
10
15
20
Iteration No.
Figure 2 Convergence Process of Q for State Feedback
GUARANTEEING OUTPUT L CONSTRAINTS
125
Hence, in the OL design the choice of W is very important and the EOL| design provides the best choice of W. Also, the cost of the OL design is very high, which implies the bound of IIu,
+11",
is large. Figures 2 and 3
provide convergence information for the discrete time
EOL| problem. Even
though this is a 10th order system, we can see that matrices Q and W converge in only 16 iterations.
Convergent Process of W
3 2.5
.... 1.5
-..~
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiii . . . .
.
.................
., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I
,,
0
5
1. . . .
10
15
20
Fig. 3 Iteration No.
Figure 3 Convergence Process of W for State Feedback B.
DISCRETE TIME FULL ORDER DYNAMIC CONTROLLER DESIGN
For the full order dynamic design, we added disturbances in the measurement channels. We assume that the L: norm of the continuous time measurement disturbance for each channel is 0.1, and all the disturbances are independent, i.e., E
w, (t)wj7 (t)dt = 0; ~' i ~: j, i, j = 1,2,3,4.
Then using the same method as in the state feedback case, we obtain the matrix bound on the outer product of closed loop input g 2 disturbance as
126
GUOMING G. ZHU AND ROBERT E. SKELTON
N =diag[200,20,O.2,0.2].
(5.9)
r =[2,2].
(5.10)
The design goals are
We use parameters defined in (5.6) (except F) to design the EOL| full order dynamic controller, as in the state feedback case, we designed two OL| controllers with W = I and W = N for comparisons, see Table 5.2 for results.
Controllers
Table 5.2 Dynamic Feedback Controller Design "OL - 1 OIL - 2 2.00 10.9
F --"2
"--2
--'2
"--2
U,
U2
.o
2.00 2.00
2.00 4.57
2.00 2.00
2.00 2.00
2.00 2.00
1.68 x 10 S 1.25 x 105 1.60x 102 5.25x 10 4.14x10 1.98x 10
W
!
Q
EOL
.0
r2.49x 10"
L
o
0
!
7.14X 10'.
N(see (5.9)) "141 0" 0
43.2
(5.11)) 28.1
0
0
19.0
Jo,~.or J eo,_~
*
2.12x102 2.90x 105 6.12x10 Iteration # 10 35 25 See the notion for Table 5.1 but the bounds becomes [ 10.9192 3.6787].
The Wor in Table 5.2 is
I 2.3039 -0. 0298 W~ = / 0 . 0 3 6 4
L-0.0032
-0.0298
0.0364
-0.0032]
0.4297
-0. 0007
0.0162 | .
-0.0007
0.0146
-0.0044 /
0.0162
-0.0044
0.0694.]
(5.11)
As in the state feedback case, the full order dynamic E O L design provides the lower design cost, see Tables 5.2. Unlike the state feedback case, the results of the O/., design with W = N are no longer close to the EOI., design, around twice the difference in the design costs J o,.. and J~.o,... Also Figure 4 shows that the pole placements for the different designs are quite different.
GUARANTEEING
OUTPUT
L
CONSTRAINTS
127
Pole Placement of EOL-inf and OL-inf
1 0.8 0.6 0.4
0"20
'
~
:
I e
-0.2 -0.4 ............................ i.............................. !.............................. i................... .t ....... -0.6 -0.8
9
-1 -1
-0.5
t
0
.
x
0.5
1
Figure 4 Closed Loop Pole Positions for Full Order Feedback Hence, the example shows that in the full order dynamic controller design, the choice of weight matrix W by the E O L algorithm is much more important than in state feedback case. In the full order dynamic OL| design the maximal accuracy bounds are available. In the discrete case, X~ satisfies
X,x = ApX,, Ar -
pX, , Mr
+ M pX, , M r- )' , M X,,A r + DpWDr9
(5.12)
and the achievable e: (i = 1 2) in (4.24) are i
"8[C X,x Cr, ]. traceNW -'" i - 1,2 9
9
(5.13)
where W = block diag[W,V]. Hence the choice of W in the OL| design will directly effect the feasibility of the O L design. When we chose W = I in the O L design in Table 5.2, it turns out that the design goal is not achievable (there are no feasible solutions). While the EOI., design does not have this problem because the algorithm will choose the weight matrix W to make constraints feasible. Also from numerical experience, the controller "gain" (cost of the E O L ) will increase as the g bounds decrease, which is consistent with the OL case. .a
128
G U O M I N G G. ZHU AND ROBERT E. SKELTON
Table 5.2 shows that the convergence rate of the OL and EOL design is very close (even though EOL. is iterating on two Riccati, while OL is iterating on one). From Figures 5 and 6 we can see that, even though this is a 20th order system, the algorithm converges in only 26 iterations.
Convergent Process of q I & q 2 _
80
.~.... ]---]
60 e,q o"
Solid line: .(:11
TD he
.....
nei q2 ...........
40
o"
.....::ii:!iiiil
20 0
ii
10
0
20
30
Fig. 5 Iteration No. Figure 5 Convergence Process of Q for Full Order Feedback
Convergent Process of W
2.6 2.4 2.2
2
1.8 ................. ,
1.6 0
i ..................................... |
10
20
30
Fig. 6 Iteration No.
Figure 6 Convergence Process of W for Full Order Feedback
GUARANTEEING OUTPUT L CONSTRAINTS
129
VI. C O N C L U S I O N S The paper presents a methodology to design measurement or dynamic controllers that guarantee output L bounds while the input L2 disturbances have an outer product which is bounded by a given constant matrix N, while minimizing a weighted summation of the L upper bound for each control channel. The feasibility of the E O L algorithm has been shown under the assumption of convergence. Finally, an example is presented for 2 special cases: state feedback and full order dynamic controller design, those example show that the EOL| design uses much less control than O L designs, given similar L2 bounds on inputs. Also the algorithms converge fast and smoothly (converges in 16 iterations for state feedback design and 26 for full order dynamic design for a 20th order system). In fact, the new E O L algorithm converges as fast as the old OL| algorithm even though the E O L algorithm iterates on two Riccati equations instead of one. Finally, inequality constraints which are not achievable using an O L algorithm might be achievable with the new E O L algorithm. The basic reason for this phenomenon is that more information about the disturbance environment is used in the E O L , where an upper bound on the L2 norm of each disturbance w~(.) is known. Only the upper bound of the L~ norm of the D
a
me ~
o
vector w(.)
utilized in the O L design, making it too qv~
-
-
i=o
conservative.
VII. A P P E N D I X A The system matrices for discrete system (5.2) in the example are Ap = block diag[ A~ , A 2 .... , A~ ],
(A.1)
where
=[0.99880.0500], [0.9801
-0.7944];,
A~ L-0-0500
0.9987j" A~ = -0.7944
0.9794 J
(A.2a)
130
A3 =
GUOMING G. ZHU AND ROBERT E. SKELTON
[0.9905
-3.9101 =[ 0.3186 A, 1-0.0377
Mr= P
-3.9101]; A, = [0.6973 0.8983 -23.5535] 0.3076 jr
[.-11.4410
0
0.9877
0.1564
0
0
O.3090
- 1.9021
0
0
=.08910
-1.3620
0
0
-.5878
3.2361
0
0
0.7071
3.5355
P
=D
P
=
(A.2b)
0.6916 J" (A.2c)
0
n
-11.4410],,
"C
7p
0.9877
0.1564
0.3090
-1.9021
-0.8910
-1.3620
-0.5878
3.2361
0.7071
3.5355
=
0.0012
0.0002
0.0494
0.0078
0.0004
-0.0024
0.0153
-0.0944
-0.0011
-0.0017
-0.0430
--0.0567
-0.0007
0.0038
-0.0263
0.1446
0.0008
0.0039
0.0266
0.1332
(A.3)
(A.4)
GUARANTEEING OUTPUT L CONSTRAINTS
131
VIII. R E F E R E N C E S 1.
C. Hsieh, J. H. Kim, G. Zhu, K. Liu, and R. E. Skelton, "An Iterative Algorithm Combining Model Reduction and Control Design," Proceeding of American Control Conference, 1990. 2. C. Hsieh, "Control of Second Order Information for Linear Systems," Ph.D. Dissertation, Purdue University, W. Lafayatte, IN, 1990. 3. G. Zhu, and R. E. Skelton, "Mixed L2 and L Problems by Weight Selection in Quadratic Optimal Control," Int. J. Contr., Vol. 33, No. 5. J. Cont. 1991. 4. D . A . Wilson, "Convolution and Hankel Operator Norms for Linear System," IEEE Trans. Auto. Contr., Vol. AC-34, No. 1, 1989. 5. G. Zhu, M. Corless, and R. E. Skelton, "Robusmess of Covariance Controllers," Allerton Conf., Monticello, IL, 1989. 6. R . E . Skelton, and G. Zhu, "Optimal L Bounds for Disturbance Robustness," IEEE Trans. Auto. Contr., Vol. AC-37, No. 5, 1992. 7. G. Zhu, and R. E. Skelton, '~Robust Discrete Controllers Guaranteeing g 2 and g| performances," IEEE Trans. Auto. Contr., Vol. AC-37, No. 5, 1992. 8. R.E. Skelton, Dynamics System Control, Wiley, New York, 1988. 9. M. Corless, G. Zhu, and R. E. Skelton, "Robust Properties of Covariance Controllers," In Proceedings of the IEEE Control and Decision Conference, Tampa, 1989. 10. G. Zhu, M. A. Rotea, and R. E. Skelton, "A Feasible Convergent Algorithm for Output Covariance Constraint Problem," In the Proceedings of 1993 American Control Conference, San Francisco, CA., 1993.
This Page Intentionally Left Blank
Techniques of Analysis and Robust Control via Zero-Placement of Periodically Compensated Discrete-Time Plants Sarit K. Das P. K. R a j a g o p a l a n 1 D e p a r t m e n t of E l e c t r i c a l E n g i n e e r i n g I n d i a n I n s t i t u t e of T e c h n o l o g y K h a r a g p u r , I N D I A 721302
I
Introduction
G i v e n a s i n g l e - i n p u t - s i n g l e - o u t p u t ( S I S O ) , linear, d i s c r e t e , t i m e inv a r i a n t ( L D T I ) p l a n t G(z) w i t h a v a r i a b l e gain K a s s o c i a t e d w i t h it, t h e s t a n d a r d c o m p e n s a t i o n p r o b l e m is to o b t a i n a c l o s e d - l o o p c o n t r o l l e r C (see F i g . l ) such t h a t b e t t e r s t a b i l i t y a n d p e r f o r m a n c e of t h e s y s t e m are a c h i e v e d . T h e case of C' b e i n g a L D T I t r a n s f e r f u n c t i o n h a s t h o r o u g h l y b e e n s t u d i e d in l i t e r a t u r e , a n d it is k n o w n t h a t a f u n d a m e n t a l l i m i t a t i o n of s u c h cont r o l l e r s is t h a t t h e y c a n n o t a l t e r t h e o p e n - l o o p zeros [1]. As a c o n s e q u e n c e , for p l a n t s h a v i n g n o n - m i n i m u m - p h a s e ( N M P ) zeros such c o n t r o l l e r s c a n not always provide satisfactory gain-margin (GM) and disturbance rejection compensation. In t h i s c h a p t e r we c o n s i d e r C' to be a l i n e a r , d i s c r e t e a n d p e r i o d i c cont r o l l e r , a n d s h o w h o w s u c h c o n t r o l l e r s c a n be u s e d to r e l o c a t e t h e s y s t e m zeros l e a d i n g t o b e t t e r G M c o m p e n s a t i o n . ( F o r t h e p u r p o s e of d i s t u r b a n c e r e j e c t i o n , h o w e v e r , t h e y h a v e b e e n s h o w n to b e of no a d d i t i o n a l a d v a n t a g e
[21131.) aPresently with the Electronics and Communication Eng.
Dept., S.R.M.
Eng.
College, Kattankulathur, T.N., India 603203. CONTROLAND DYNAMICSYSTEMS,VOL. 74 Copyright 9 1996by AcademicPress, Inc. All rights of reproductionin any form reserved.
133
134
SARIT K. DAS AND E K. RAJAGOPALAN
Jcompensator I
J
plant
Figure 1- T h e closed loop control scheme
A M - p e r i o d i c i n p u t - o u t p u t m a p g is one t h a t satisfies the relation gz
-M
=
z
-M
9
(1)
where z -1 is the unit delay operator. Now, the questions t h a t first arise as one c o n t e m p l a t e s using such a M - p e r i o d i c C are: 1. In w h a t form should C be assumed? 2. How should such a s y s t e m be analysed? T h e s e questions are, however, inter-related and may be asked in t h e reverse order as well. As a m a t t e r of fact, the 2nd question was a d d r e s s e d first in [2] wherein it was shown t h a t a SISO, M - p e r i o d i c m a p 9 can be lifted isomorphically to a M - i n p u t M - o u t p u t time invariant form which, in t u r n , can be represented as a proper, L D T I , M x M transfer m a t r i x (~(z) with the a d d i t i o n a l causality p r o p e r t y t h a t ( ~ ( ~ ) is lower triangular. Consequently, in [2] t h e periodic c o m p e n s a t i o n problem was t r a n s f o r m e d into a m u l t i - i n p u t m u l t i - o u t p u t ( M I M O ) L D T I problem by (i) viewing the given L D T I plant as a M - p e r i o d i c one and o b t a i n i n g the c o r r e s p o n d i n g M x M L D T I transfer m a t r i x , and then (ii) assuming the periodic controller to be a M x M L D T I transfer m a t r i x C whose elements are to be o b t a i n e d using s t a n d a r d M I M O m e t h o d o l o g i e s to achieve the required closed loop characteristics. T h i s approach, however, has the following s h o r t c o m i n g s : 1. T h e causality condition t h a t C(c~) be lower t r i a n g u l a r is not automatically ensured. So far in l i t e r a t u r e this p r o b l e m has been t a k e n care of either by ad hoc m e a n s (as in [2]) or by c o n s t r a i n e d solution techniques (e.g. linear p r o g r a m m i n g , as in [4]). 2. T h e periodic controllers t h u s o b t a i n e d are not g u a r a n t e e d to be minimal (see A p p e n d i x A). In [5], on t h e o t h e r h a n d , a m e t h o d based on t h e F l o q u e t t h e o r y [6] was developed for o b t a i n i n g t h e characteristic e q u a t i o n (CE) of periodic coefficient difference e q u a t i o n s directly. Further, t h e controller was a s s u m e d in a SISO, causal, ruth order, 2-periodic form having the m a x i m u m possible
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
135
degrees of freedom. T h e stability of the closed loop s y s t e m was t h e n analysed by o b t a i n i n g the CE c o r r e s p o n d i n g to its difference equation. This approach t h u s p e r m i t s one to m a n i p u l a t e the form of the closed loop C E by choosing the controller p a r a m e t e r s appropriately, b u t does not yield the i n p u t - o u t p u t relations. In this c h a p t e r we begin by assuming a M - p e r i o d i c s t r u c t u r e h a v i n g the m a x i m u m possible degrees of freedom for C. We t h e n o b t a i n t h e lifted L D T I transfer m a t r i x c o r r e s p o n d i n g to the same using which the closed loop s y s t e m can be analysed employing L D T I methodologies. We also show how the CE of the s y s t e m can be o b t a i n e d directly following the F l o q u e t t h e o r y approach. Next we investigate the zero p l a c e m e n t capability of periodic controllers. In this context we first discuss zero placement in the conventional sense as done in [7], and t h e n show following [5] how zero p l a c e m e n t in t h e sense of modifying the high loop-gain roots of the CE can be achieved using 2-periodic controllers. Moreover we e x t e n d the l a t t e r a p p r o a c h to o b t a i n some results regarding the zero p l a c e m e n t capability of general M - p e r i o d i c controllers. Finally we discuss the aspect of improved GM c o m p e n s a t i o n of L D T I plants using periodic controllers. While L D T I controllers can provide only a finite GM c o m p e n s a t i o n to u n s t a b l e N M P plants, it is easy to see t h a t periodic controllers can, by v i r t u e of their zero p l a c e m e n t capability, provide infinite G M c o m p e n s a t i o n to such plants if t h e y are bicausal [5][7]. Now, a l t h o u g h for strictly proper plants infinite GM c o m p e n s a t i o n is not possible, it has been observed in l i t e r a t u r e t h r o u g h isolated e x a m p l e s t h a t in some cases periodic controllers can provide b e t t e r GM c o m p e n s a t i o n t h a n L D T I ones. However, no definite result is as yet available in this regard. In this context we present some results t h a t allow one to provide o p t i m u m GM c o m p e n s a t i o n to strictly p r o p e r plants via the zero p l a c e m e n t approach.
II
Controller Configuration
Unlike the L D T I case, the form in which periodic controllers should be sought is not well-standardized, and we will have to m a k e a suitable choice for the same. Now, keeping in view the facts that the controller (i) should be realizable, and (ii) should have the m a x i m u m possible degrees of freedom so that minimal order designs could be achieved, we make,
136
SARIT K. DAS AND E K. RAJAGOPALAN
following essentially [5], t h e following ruth o r d e r M - p e r i o d i c choice: yi(N
+ l) =
yi+l(N),
~--1l
i=
ym(N) = L(N)e(N) - ~
0,1,.
. . , (m -
1)
C,(N)y,(N)
! (2)
i=0
u(N) = ~D,(N)y,(N)
"
I
i--O
w h e r e e ( N ) a n d u ( N ) are, respectively, t h e i n p u t a n d t h e o u t p u t of t h e controller, a n d C i ( N ) and D i ( N ) are, respectively, t h e feedback a n d feedf o r w a r d gains t h a t are M - p e r i o d i c . N o t e t h a t to specify Ci or Di one has to give t h e i r values at t h e t i m e i n s t a n t s N ---- 0, 1,. - . , ( M - 1), or, a l t e r n a t i v e l y , t h e y m a y be d e s c r i b e d in t h e f o r m of d i s c r e t e F o u r i e r series as
C , ( N ) = C i ( N + M ) = M-ik=~ ~ ~k~Vc, k,
i = 0, 1 , - . . , (m -- 1)
!
(3) Di(N) = Di(N + M) = ~
otk"dik,
i = O, 1 , . . . , m
J
k=0
where c~ := ezp(j2~r/M)
(4)
Fig.2 shows t h e closed loop s y s t e m consisting of this controller a n d t h e plant. (It m a y be n o t e d in this c o n t e x t t h a t this c o n t r o l l e r is o b t a i n e d ext e n d i n g t h e c o n t r o l l e r c a n o n i c a l L D T I s t r u c t u r e [1]. A l t e r n a t i v e b u t equivalent controllers which are periodic e x t e n s i o n s of t h e o t h e r L D T I c a n o n i c a l s t r u c t u r e s can also be o b t a i n e d . ) T h e a b o v e controller is, by c o n s t r u c t i o n , causal a n d realizable. F u r t h e r , since a n y a d d i t i o n a l M - p e r i o d i c gain i n t r o d u c e d at a n y p o i n t in it can be a b s o r b e d in t h e C i ' s a n d D i ' s (see A p p e n d i x B), it can also be said to have t h e m a x i m u m possible degrees of f r e e d o m . In fact, if t h e gain t e r m L ( N ) ~: O, VN, t h e n it can also be similarly a b s o r b e d . B u t if L ( N ) -- 0 for s o m e N , t h e n one can not do so (since t h e process of a b s o r p t i o n c a u s e s a factor L - X ( N ) to a p p e a r in t h e feedback gain t e r m s . ) T h e l a t t e r , however, is only a special case a n d will be t r e a t e d separately. H e n c e f o r t h , unless s t a t e d o t h e r w i s e , we will h a v e L ( N ) -- 1. For L D T I s y s t e m s it is often m o r e a d v a n t a g e o u s to d e a l w i t h t r a n s f e r f u n c t i o n d e s c r i p t i o n s t h a n with t i m e d o m a i n ones. Now, it is s t r a i g h t f o r w a r d to see f r o m Fig.2 t h a t t h e e q u i v a l e n t periodic coefficient ' t r a n s f e r f u n c t i o n ' c o r r e s p o n d i n g to t h e controller is
C(~,N)
[D~(N)~ ~ + D~_~(N)~ ~-~ +---+ [~
+ C~_~(N)~ ~-~ +...
D0(N)].
+ C o ( N ) ] -~
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL I I i I I !
137
! I
4
I I I I I I
*
'
i I I I I I
Yrn--1
I
:
I
'I J
: ~
1
, ,
L Co V
!
t
Di(N) L(N)
- Di(N
+ M)
Controller J Ci(N)
-- C i ( N + M )
(Nominally L(N)-
= L(N + M)
plant
I I I I I I I I
....
! I i
,,
I' I ! !
Yl
I I i I
I I l l I
1)
Figure 2: T h e M - p e r i o d i c controller and the plant
which, using eqn.(3), m a y also be written as C(z,N)
=
[qo(z) + c~Nql(z) + "'' + o~(M--1)NqM--I(z)].
(5)
[po(~) + ~ p ~ ( z ) + ... + ~(~_~).p~_~(~)l-X where qi = dm , i z '~ + d m - l , i
Z m-I
rrt--1
Po -- zm + c m - l , o Z +''" Pl = c m - l , i z m - 1 + ' ' ' + C o , i ,
+ . ' . + do, i,
i = O, 1 , . . . ,
( M - 1)
+ Co,o i= 1,2 .... ,(M--l)
I
(6)
)
H a v i n g t h u s chosen the controller our next j o b is to see how it affects t h e s y s t e m b e h a v i o u r . Unlike the L D T I case, however, t h e m e t h o d s for d e a l i n g with periodic s y s t e m s are not well known. T h e s e are d i s c u s s e d n e x t .
III
Analytical odic
Techniques
for Discrete
Peri-
Systems
As n o t e d in S e c t i o n I, two a p p r o a c h e s are available for d e a l i n g with periodic s y s t e m s . In one, periodic s y s t e m s are lifted to larger L D T I forms,
138
SARIT K. DAS AND P. K. RAJAGOPALAN
and, in t h e o t h e r , t h e C E s of p e r i o d i c coefficient difference e q u a t i o n s are obtained using the Floquet theory.
A
The
Lifting
Technique
C o n s i d e r a S I S O , c a u s a l , 2 - p e r i o d i c m a p g. L e t e ( 0 ) , e ( 1 ) , . . , b e a s e q u e n c e of i n p u t s to this m a p a n d let u(0), u ( 1 ) , . . , be t h e c o r r e s p o n d i n g s e q u e n c e of o u t p u t s . Now, collect t h e e v e n a n d o d d i n s t a n t i n p u t s a n d o u t p u t s s e p a r a t e l y a n d r e p r e s e n t t h e m in t h e t r a n s f o r m d o m a i n as oo
E~(z 2)
--
~
z-2ie(2i)
i=0 oo
z
-X
Eo(
Z2
) =
~
z - 2 i - 1 e(2i + 1)
i----0
U~(z 2) =
oo
z
u(2i
i=0 oo
z --1 Uo(z 2 )
Z --21--1 e
----
(2i + 1)
i=0
so t h a t o n e has
E(z)
--
E~(z 2).+- z - t E o ( z 2)
U(,)
=
U~(~ ~) + z -` Uo(z ~)
w h e r e E(z), Ee(z 2) a n d z-~Eo(z 2) can be identified to be t h e z - t r a n s f o r m s of, r e s p e c t i v e l y , t h e c o m p l e t e , t h e e v e n i n s t a n t a n d t h e o d d i n s t a n t i n p u t s e q u e n c e s , a n d , similarly, U(z), Ue(z 2) a n d z - l U o ( z 2) s t a n d for t h e c o m plete, t h e e v e n i n s t a n t a n d t h e o d d i n s t a n t o u t p u t s e q u e n c e s . Now it is e a s y to see t h a t t h e o n l y way t h e even a n d o d d i n s t a n t o u t p u t s can b e linked to t h e e v e n a n d o d d i n s t a n t i n p u t s is
z_XUo(z2 )
- G(z)
z_XEo(z2 )
(7)
with
:=
G11(z2 ) z _ ~G2t( z~ )
z-~Gt2(z 2) ] G22 ( ~ )
w h e r e e a c h Gij is a L D T I , p r o p e r t r a n s f e r f u n c t i o n in z ~. ( T h a t Gij are f u n c t i o n s of z 2 is o b v i o u s . Also, t h e y m u s t be t i m e i n v a r i a n t so as to satisfy t h e 2 - p e r i o d i c i t y c o n d i t i o n as follows f r o m e q n . ( 1 ) . F u r t h e r , t h e y h a v e to be p r o p e r to e n s u r e c a u s a l i t y . ) T h e m a t r i x G ( z ) is t h e n t h e lifted,
TECHNIQUES OF ANALYSISAND ROBUST CONTROL
139
LDTI, t r a n s f o r m d o m a i n r e p r e s e n t a t i o n of a general 2-periodic map. Note, however, t h a t eqn.(7) may also be written as
Uo(z 2)
] z-2G12(z2)]
]= +(z2
where [ G l l (z2)
Eo(z ~)
Consequently, one may, following [2], equivalently view C?(z2) to be the transfer m a t r i x c o r r e s p o n d i n g to 2-periodic maps. Notice t h a t (~(oo) is lower triangilar, and this is the necessary and sumcient condition for an a r b i t r a r y 2 x 2 p r o p e r transfer function to be realizable as a SISO 2-periodic system. T h e above result can be e x t e n d e d to conclude (as in [2]) t h a t a linear, causal, M - p e r i o d i c p t - i n p u t p 2 - o u t p u t m a p g can be uniquely r e p r e s e n t e d as a p : M x p t M t i m e invariant transfer m a t r i x G(z M) which has t h e p r o p e r t y t h a t G(oo) is lower triangular. Conversely, given any p~M x P t M transfer m a t r i x G(z M) with G(oo) lower triangular, one can find a unique causal p t - i n p u t p 2 - o u t p u t M - p e r i o d i c m a p g such t h a t the c o r r e s p o n d i n g C~ = G. A l t h o u g h this result has p r o m p t e d several workers [2][4][7] to a s s u m e periodic controllers as M I M O L D T I transfer matrices, it has already been p o i n t e d out in Section I t h a t such a n approach has some s h o r t c o m i n g s . We, however, have chosen the controller as shown in Fig.2, and next show how the equivalent 2 • 2 L D T I transfer m a t r i x c o r r e s p o n d i n g to the s a m e can be o b t a i n e d . L D T I Transfer Matrix corresponding to Periodic Controllers of the form of Fig.2.
For simplicity we restrict our attention to only the 2-periodic case in this section. For M -- 2, the time varying transfer function description of the controller (with L(N)-- 1) as given by eqn.(5) becomes
C ( z , N ) -- [qo(z) +
(--1)Nql(z)][Po(Z)
"4- (--1)NpI(Z)] -t
(9)
We n o w have the following result. Theorem
1 The LDT] transfer matriz corresponding to eqn.(9) is
d,(,) =
1
['~+~ '~]
(lo)
140
SARIT K. DAS AND R K. RAJAGOPALAN
with Cll = q+oPo + q o P + -- q+P'( -- qoP + + q+l Po + q ; P + -- q+P'( -- q l p+ c12 "- (q+Po -- qoP + + q+oP'l -- qoP + + q+Po -- q~P+o + q + P l -- q ~ p + ) / z c21 -" (q+o Po -- qoP+o -- q+oP'( + qoP+l -- q+Po + q~P+ + q+l P l -- q~P+l )z c22 "- q+o Po + qoP+o + q+oP'l + qoP+l -- q+Po -- q~P+o -- q + P l -- q l P + where, f o r any p o l ! l n o m i a l f ( z ) ,
f+ = f(z)
and f -
= f(--z).
Proof." C o n s i d e r first the i n p u t - o u t p u t m a p given by
(11)
y ( N ) = [q0(z) + ( - - 1 ) N q l ( z ) ] z ( N ) Now, split up qo(z) into its even a n d o d d p a r t s as qo,e(z2) z
-I
=
(q+ + q o ) / 2
)
=
(qo+ - q o ) / 2
qx,e(z2)
=
(q+l + q~ ) / 2
ql,o(z2)
=
(q+l
qo, o(
Z2
Similarly, o b t a i n
--1
-- q : ) / 2
F u r t h e r , express t h e z - t r a n s f o r m s of z ( N ) a n d y ( N ) t r a n s f o r m s of t h e i r even a n d o d d i n s t a n t s e q u e n c e s as
x(~) Y(z)
= =
in t e r m s of t h e z-
x,(~ 2) + ~-XXo(:~) Y,(z 2) + z-Iyo(Z 2)
T h e n it is s t r a i g h t f o r w a r d to o b t a i n from eqn.(11) t h a t Y~(:~)
=
(1/2)[q + + qo + q+ + ql-]Xe(z2) + ( 1 / 2 ) [ q 0+ - q 0 -
-"
+ q+l - q ? ] z - l X o ( z ~ )
(1/2)[qo+ - qo - q+ + ql"]Xe(z2) + (1/2)[q0+ + qo - ql+ - q l ] Z - l X o ( z 2 )
which can also be w r i t t e n as
[ yo(~ y~(~2))] = ~1 [ X~(~') ] Xo(Z 2)
where
1 [ qo+ + q o + q + -t- qxz(q+o -- qo -- q+ + q l )
(~1 := ~
~ - '(q~+o- q~ + q +l - q { ) ] +
qo
- q+~ -
qF
(12)
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
141
Now defining
G~
i
[
:= ~
p+ +po +,~+ + p r ~ b o+ - po - p+ + p[]
~ - ~ b 0 + - po + p,+ - pT] ] ] po+ + po - p~+ - p[
it follows obviously f r o m eqn.(9) t h a t the L D T I t r a n s f e r m a t r i x c o r r e s p o n d ing to t h e c o n t r o l l e r is 0 = ~i~ i which, on simplification, leads to eqn.(lO). Since a L D T I p l a n t is also a 2-periodic one, it would h a v e a 2 x 2 r e p r e s e n t a t i o n as well.
Corollary
1.1 For a L D T I transfer.function C ( z ) = q(z)/p(z) the equivalent 2 x 2 representation is
(~_
1 [ q+p- + q - p + 2p+pz(q+p - -- q-p+)
z-l(q+p---q-p+) ] q+p- -4- q-p+
Proof." Follows directly from eqn.(10) by s e t t i n g Pl = ql = 0.
(13) O
N e x t we bring o u t two special cases of t h e g e n e r a l 2-periodic controller. i) Even Output controller Referring to eqn.(9) it is seen t h a t if qx (z) -- qo(z) (or, w h a t is t h e s a m e thing, if Di(N) = 0 for o d d N ) t h e n t h e controller o u t p u t will be zero at t h e o d d i n s t a n t s . U n d e r this c o n d i t i o n we h a v e from eqn.(10) t h a t
1
[ q+po +qop +-
(~ ---- P+Po -- P+P'I
q+p'~--qop +
z-1(q+oPo -- qo p+ + q+P-1 -- qo p+)
0 ]T 0
(14)
N o t e t h a t such a controller m a y also be i m p l e m e n t e d as a c o m p o s i t e s y s t e m consisting of a t i m e i n v a r i a n t u n i t 2q(z)/p(z) followed by an even i n s t a n t 2 T - s a m p l e r . Using C o r o l l a r y 1.1 t h e 2 x 2 t r a n s f e r m a t r i x c o r r e s p o n d i n g to t h e c o m p o s i t e s y s t e m can be o b t a i n e d to be
~1
Clearly, r
--
1 [ 1 0 ] [ q+p - + q - p + z-l(q+p - - - q - p + ) ] p+p 0 0 . . . . . . 1 q+P- +q-P+ z-X(q+P- --q-P+) ] p+p 0 0
(15)
b e c o m e s s a m e as C if one chooses
.+
=
p(z)
=
.o+po-p~+p? ),
q+
=
qC z )
=
q+o p o - qo p +~ J
(16)
142
SARIT K. DAS AND E K. RAJAGOPALAN
Note, however, t h a t this i m p l e m e n t a t i o n is of o r d e r 2rn, rn being t h e o r d e r of t h e original controller. Similarly, t h e choice of ql(z) -- -qo(z) leads to an odd output c o n t r o l l e r which can be i m p l e m e n t e d using an o d d i n s t a n t 2 T - s a m p l e r . ii) Even Input controller Such a c o n t r o l l e r (one t h a t rejects o d d i n s t a n t i n p u t s ) can be o b t a i n e d by h a v i n g a L D T I t r a n s f e r f u n c t i o n 2q(z)/p(z) p r e c e d e d by an even i n s t a n t 2 T - s a m p l e r . T h e c o r r e s p o n d i n g 2 x 2 t r a n s f e r m a t r i x is
C- =
1 [ q+ p - + q-p+ p+pz(q+p - - - q - p + )
0 ]
(17)
0
Now, in t e r m s of Fig.2 t h e s a m e c o r r e s p o n d s to choosing L ( N ) as an even i n s t a n t s a m p l e r and s e t t i n g Pl
p+ q+
=
0 (is)
---- q+Po "+- q~P+
)
An odd input c o n t r o l l e r m a y also similarly be o b t a i n e d . 2
Closed Loop Characteristic
Equation
o f 2-Periodic S y s t e m s
Having shown how t h e 2-periodic c o n t r o l l e r as well as t h e p l a n t can be d e s c r i b e d by 2 x 2 L D T I t r a n s f e r matrices, we are now in a p o s i t i o n to a n a l y s e t h e s t a b i l i t y as well as the r e s p o n s e of t h e overall s y s t e m using L D T I m e t h o d o l o g i e s . In this c h a p t e r , however, we are c o n c e r n e d o n l y with stability, and hence o b t a i n n e x t t h e C E of t h e s y s t e m for t h e g e n e r a l and t he special c o n t r o l l e r forms using t h e well known r e l a t i o n A : d e t [ I + KGC]
(19)
where G c o r r e s p o n d s to t h e p l a n t G(z) -- b(z)/a(z) and, following C o r o l l a r y 1.1, is given by
(~=
1 2a + a-
[ b+a-+b-a + z(b + a - - b-a +)
z-l(b+a--b-a+) ] b+ a- -4- b-a +
and C is t h e 2 x 2 t r a n s f e r m a t r i x c o r r e s p o n d i n g to t h e c o n t r o l l e r chosen. i) W i t h C chosen in t h e g e n e r a l form as given by eqn.(10), t h e C E b e c o m e s
=
~ + " - (p0 + p o - p,+pF)
+ K[ b+a-(q+po -- q~P+l) + b-a+(qoP+o -- q+lP'~)] -+- K2b+ b- (q+ qo - q+ q~ )
(20)
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
143
ii) For an even o u t p u t c o n t r o l l e r which r e s u l t s if one sets q+ = q+, one obtains _
_
+ K[b+a-(q+oPo -- q~P+l) + b-a+(qoP+o
q+lP'l)]
(21)
N o t e t h a t this C E is s a m e as t h e one given by e q n . ( 2 0 ) with t h e K 2 - t e r m a b s e n t . N o t e also t h a t t h e s a m e C E r e s u l t s if one c h o o s e s q+ as q+ = - q + , q0 o r - - q o " iii) For t h e even i n p u t c o n t r o l l e r t h e C is given by e q n . ( 1 7 ) w i t h p a n d q a s in e q n . ( 1 8 ) , a n d t h e C E b e c o m e s
A -- a+a-P+oPo + K[b+a-(q+op o + q ~ p + o ) + b - a + ( q o P + o
-I-q+lpo)]
(22)
T h e r e s u l t s o b t a i n e d in this s e c t i o n can also be e x t e n d e d t o 3- or h i g h e r p e r i o d i c s y s t e m s . We, however, do n o t use e q n . ( 1 9 ) to e x p l i c i t l y o b t a i n t h e C E s of such s y s t e m s since, as is s h o w n n e x t , t h e s a m e can b e o b t a i n e d in a m o r e d i r e c t f a s h i o n using t h e F l o q u e t t h e o r y .
B
Analysis using Floquet Theory
T h i s m e t h o d , d e v e l o p e d in [5], yields t h e C E of a p e r i o d i c coefficient difference e q u a t i o n in t e r m s of t h e d i s c r e t e F o u r i e r coefficients of t h e periodic p a r a m e t e r s . T h e p e r i o d i c c o n t r o l l e r b e i n g r e p r e s e n t a b l e as in e q n . ( 5 ) , it is n o t difficult to see t h a t t h e difference e q u a t i o n of t h e overall s y s t e m of Fig.2 in t e r m s of t h e v a r i a b l e Y0 is
[a(~){p0(z) + ~ p ~ ( ~ ) + .-. + ~ ( ~ - ~ ) N p ~ _ ~ ( z ) } + K b ( z ) { q o ( z ) + ~ N q l ( z ) + . . . + ot(M--1)NqM_~(Z)}]yo(N ) "-- O
(23)
T h i s e q u a t i o n is, h o w e v e r , i n c o n v e n i e n t for a n a l y s i s since t h e t i m e v a r y i n g t e r m s o c c u r in b e t w e e n o p e r a t o r s . To avoid this t h e following r e s u l t is useful. Lemma
1 For any p o l y n o m i a l f ( z )
C~kN f ( Z ) =
in the operator z
,f(otM-kz)oekN,
k = 1, 2 , . . . , ( M -
1)
where ~ is as given by eqn.(4).
Proof: N o t e f r o m e q n . ( 4 ) t h a t for a M - p e r i o d i c s y s t e m ~M = 1. W e t h e n h a v e for any i n t e g e r i t h a t [~kNzl]z(N)
=
~ - - k i ~ k N + l e i z ( N + i)
.._
ol(M--k)izi { ~ k N z ( N ) }
=
[{~"-~z}'~],(N)
144
SARIT K. DAS AND E K. RAJAGOPALAN
Hence the result follows. Using this result eqn.(23) m a y be r e w r i t t e n as
M_,
]
(24)
Now, according to t h e classical F l o q u e t t h e o r y [6], a c o n t i n u o u s t i m e differential e q u a t i o n with periodic coefficients has solutions of the f o r m z(t) -exp(At)h(t) with h(t) = h(t + Te), Tp being t h e period of t h e periodic par a m e t e r s . For a M - p e r i o d i c discrete t i m e s y s t e m we have Tp -- M T , where T is the intrinsic s a m p l i n g period of the system. W i t h o u t any loss of generality we a s s u m e T = 1. T h e n , replacing t by N and defining 7 := exp(A), the discrete version of the F l o q u e t solution of eqn.(24) b e c o m e s
(25)
yo(N) - 7 ~ h ( N ) with h ( N ) -- h ( N + M ) Further, let the discrete Fourier series for h ( N ) be
h ( N ) = ho +crlVhx + tr2JVh~ + . . . + ot(M-1)NhM_ 1
(26)
We t h e n have t h e following result. Theorem
2 For systems governed by difference equations of the form oJ
det[a(~iT)pM+i-j(~JT) -}" Kb(~iT)qM+i-j(oJT)]i,jfo,t ..... (M-X) -- 0 (27) where c~ is as given b y e q n . ( ~ ) , P M + I = P - M + i qi. Writing in full, for M -- 2 the CE is a ( 7 ) p 0 ( 7 ) + Kb(7)qo(7) a ( - 7 ) p x (7) + K b ( - 7 ) q x (7)
=
Pl and
qM+i
= q-M+i
a(7)px (--7) + K b ( 7 ) q x ( - 7 ) a(-7)p0(-7) + Kb(-7)qo(-7)
=
=0
Similarly, .for M -- 3 eqn.(f7) becomes
~(~)p0(~) + Xb(~)qo(~) a(crT)pl (7) + Kb(~7)ql (7)
a(7)p~(c~7) + Kb(7)q2(~7) a ( 7 ) p x ( c ~ 7 ) + Kb(7)ql(~27) a(trT)p0(trT) + Kb(trT)qo(trT)
~(~)p2(~) a ( c ~ 7 ) p 2 ( 7 ) + Kb(~27)q2(7)
+ Kb(~)q2(~)
=0
a(c~27)Px (~7) + Kb(~27)ql ( a T )
a(a27)po(a~7) + Kb(a27)qo(tr27) Proof: P u t eqns.(25) and (26) in (24), then collect t h e coefficients of t h e a i N - t e r m s , i = 0, 1 , . . . , ( M 1), a n d individually e q u a t e t h e m t o zero.
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
145
Since for any p o l y n o m i a l f ( z ) in t h e o p e r a t o r z, f ( z ) 7 N -- f(3,)'y N, t h e set of e q u a t i o n s t h u s o b t a i n e d m a y be w r i t t e n as M-1
[a(criT)pk(crM+i-kT) + K b ( c r i T ) q + k ( c r M + i - k T ) ] h i _ k = O, k---O
i-- 0,1,...,(M-
1)
Since hM+i = h - M + i = hi, this set of e q u a t i o n s can also be e x p r e s s e d in the m a t r i x form as FH=0 F being t h e M x M m a t r i x shown in eqn.(27) an d H = [ho hl . . . hM_x] z'. T h e coefficients hi are, however, a r b i t r a r y , a n d hence t h e C E b e c o m e s as given by eqn.(27). D To verify t h e result check t h a t for M = 2 eqn.(27) can be e x p a n d e d into p o l y n o m i a l form to lead to t h e s a m e C E as given by eqn.(20) which was o b t a i n e d using t h e lifting technique. C o m p a r i n g eqns.(19) and (27) it is seen t h a t t h e l a t t e r yields t h e C E in a m o r e direct fashion t h a n t h e former. Hence, so far as s t a b i l i t y a n a l y s i s is c o n c e r n e d , t h e F l o q u e t t h e o r y a p p r o a c h is m o r e c o n v e n i e n t t h a n t h e lifting technique. Unlike t h e l a t t e r a p p r o a c h , however, t h e f o r m e r does n o t have t h e c a p a b i l i t y of yielding t h e i n p u t - o u t p u t r e l a t i o n as well. For s y s t e m s which c a n n o t be d e s c r i b e d by a difference e q u a t i o n of t h e form of eqn.(24) (as would be t h e case for M I M O p er i o d i c s y s t e m s ) t h e following m o r e g e n e r a l result m a y be utilised to o b t a i n t h e CE. Theorem
3 For s y s t e m s described in the state variable f o r m as X(N
+ 1) -- A ( N ) X ( N )
with A ( N )
-- A ( N + M )
the C E is Ao - 7 I A~
AM-1 Ao - ( ~ ' r ) I
AM-2 AM-1
"'" ""
A1 A2
AM-1
AM-2
AM-3
""
Ao-- (c~M-l'/)I
--0
where c~ is as given by eqn.(4) and A i s t a n d s f o r the ith coefficient o f the discrete F o u r i e r series f o r the e l e m e n t s of the m a t r i z A , i.e., A(N)
= A s + c~N A1 + c~2N A2 + " " + c~( M - x ) N A M - 1
Proof: T h i s result is easily o b t a i n e d p r o c e e d i n g as in T h e o r e m 2.
[-!
It is i n t e r e s t i n g to n o t e in this c o n t e x t t h a t t h e finite d i m e n s i o n a l det e r m i n a n t s o b t a i n e d in T h e o r e m s 2 and 3 become, for p e r i o d i c c o n t i n u o u s t i m e s y s t e m s , infinite d e t e r m i n a n t s [8].
146
SARIT K. DAS AND R K. RAJAGOPALAN
IV
Salient Features of the CE cally Compensated Systems
of Periodi-
T h e stability of a s y s t e m being decided by t h e roots of its CE, it is necessary to s t u d y carefully the salient features of the C E of periodically c o m p e n s a t e d s y s t e m s in order to obtain an u n d e r s t a n d i n g of their stability characteristics. From eqn.(20), the following observations regarding the C E of 2-periodically c o m p e n s a t e d s y s t e m s can easily be made: 1. T h e y are even p o l y n o m i a l s in z. 2. T h e y are 2nd degree polynomials in the loop gain K. 3. T h e coefficient of the K ~
contains the fixed factor a + a - .
4. T h e coefficient of the K 2 - t e r m contains the fixed factor b+ b - . 5. T h e coefficient of the K 1-term can be arbitrarily fixed provided b o t h (b +, b - ) and (a +, a - ) are m u t u a l l y coprime pairs. To u n d e r s t a n d the properties of t h e CE for the general M - p e r i o d i c case, we first obtain, e x p a n d i n g eqn.(27), the CE for M -- 3 as 3
'~
= F_, K ' / , ( ~ )
(28)
i--O
where
-~
k---O
---
i---O
2
,=0
f2
=
fa
-'-
-- po(c~ i z)ql (c~i+l z)p,(c~i+2 z) -- qo(c~i+2z)pl (cr i z)p2(oli+lz) -- p0(c~/+1 z)pl (c~i+2z)q2(c~ i z)] s a m e as fl with pi(z) replaced by qi(c~z), qi(z) replaced by pi(o~z), a(z) replaced by b(otz) and b(z) replaced by a(c~z). s a m e as fo with t h e p's replaced by q's and a by b.
Based on a careful study of eqn.(28) we now m a k e the following obserrations regarding the Properties of the C E s of M - p e r i o d i c systems:
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
147
1. T h e C E is a M t h d e g r e e p o l y n o m i a l in K of t h e f o r m M
(29)
~_~ K ' f i ( z M ) "- 0 i--.-O
2. fi, t h e coefficient of t h e K i - t e r m , i -- 0, 1 , . . . , M , is a p o l y n o m i a l in z M of d e g r e e e, : - m -I- n - Z + { i ( n - r ) / M } (30) w h e r e n a n d r are, r e s p e c t i v e l y , t h e d e g r e e s of t h e a ( z ) a n d t h e b ( z ) p o l y n o m i a l s , m is t h e c o n t r o l l e r o r d e r , a n d 2"+ is t h e u p p e r i n t e g e r or ceiling o p e r a t o r . [For M --- 3, for e x a m p l e , 90 = Z [ { 3 n + 3 m } / 3 ] = n + m --" :Z[{(2n -t- r) + 3 m } / 3 ] -- n -+- m -- Z + [ ( n -- r ) / 3 ] 02 -- ~ [ { ( n + 2 r ) + 3 m } / 3 ] -- n -t- m -- ~ + [ 2 ( n -- r ) / 3 ] 03 -" ~ [ { 3 r + 3 m } / 3 ] -- r + m
01
w h e r e Z is t h e u s u a l i n t e g e r (or, floor) o p e r a t o r . ] 3. f0 c a n b e e x p r e s s e d as fo -- f o o f m , w h e r e f01 is a n a r b i t r a r y r u t h d e g r e e p o l y n o m i a l ( w h i c h d e p e n d s on t h e c h o i c e of t h e c o n t r o l l e r p a rameters) but n
S00 = ~ ( ~ ) ~ ( ~ ) , , ,
a(~-~)
= II(~ ~ - ~Y) iffil
w h e r e ai a r e t h e p l a n t p o l e s , i.e. z e r o s of a ( z ) . 4. f M c a n b e e x p r e s s e d as f M -~ f M o f M 1 , w h e r e f M 1 is a n a r b i t r a r y r u t h d e g r e e p o l y n o m i a l ( w h i c h d e p e n d s on t h e c h o i c e of t h e c o n t r o l l e r parameters) but
fMo = b(z)b(~z) . . . b(~M-lz)
-- H ( z
M -- biM )
i--1
w h e r e bi a r e t h e finite p l a n t zeros, i.e. z e r o s of b ( z ) .
5. f i , i ~ 0 or M , a r e c o m p l e t e l y a r b i t r a r y p o l y n o m i a l s of d e g r e e 0i p r o v i d e d t h a t t h e f o l l o w i n g c o n d i t i o n is satisfied: Condition
1 I f 7 is a pole ( o r zero) of the p l a n t t h e n n o n e of the
pole (or zero).
148
SARIT K. DAS AND E K. RAJAGOPALAN
6. If, h o w e v e r , C o n d i t i o n 1 is n o t satisfied, t h e n fi, i ~ 0 or M , will n o t all be a r b i t r a r y as given by t h e following rule: If, for a n y 7, M~ of t h e M p o i n t s 7 e x p ( j 2 ~ r k / M ) , k = 0, 1 , . . . , ( M - 1), are poles of t h e p l a n t , t h e n fi, 0 _< i _< (M1 - 1), will h a v e a fixed f a c t o r (z M - 7 M ) M I - 1 a s s o c i a t e d w i t h it. A g a i n , iffor a n y 7, M2 of t h e p o i n t s 7 e x p ( j 2 7 c k / M ) , k = 0, 1 , . . . , ( M 1), are finite zeros of t h e p l a n t , t h e n fi, ( M - M2 + 1) <_ i <_ M , will h a v e a fixed f a c t o r (z M _ 7 M ) M 2 - I+M a s s o c i a t e d w i t h it. [For e x a m p l e , s u p p o s e t h a t t h e given p l a n t is G ( z ) -- 1 / ( z - 71 )(z + 0.572 + j O . 8 6 6 7 2 ) ( z + 0.572 -- j 0 . 8 6 6 7 2 ) T h e n , for M -- 3, f l will h a v e a fixed f a c t o r (z 3 + 7~) a s s o c i a t e d w i t h it.] It is o b v i o u s t h a t in a l m o s t all cases C o n d i t i o n i will b e satisfied. In a n y case, for a given p r o b l e m M c a n b e so c h o s e n t h a t it is satisfied. In g e n e r a l , t h e n , o n e m a y say t h a t a l t h o u g h t h e coefficients of f0 a n d f M a r e n o t fully a r b i t r a r y , t h o s e of f/, i ---- 1, 2 , . . . , ( M 1), are. S u c h b e i n g t h e case, o n e w o u l d n e x t like to e x a m i n e w h a t is t h e m i n i m u m o r d e r of t h e c o n t r o l l e r n e c e s s a r y for t h e p u r p o s e of a r b i t r a r y a s s i g n m e n t of t h e C E coefficients. 4 Referring to the characteristic equation (~9) of periodically compensated systems and assuming that Condition 1 is satisfied, in order to be able to assign arbitrarily all the c o e ~ c i e n t s of fi, i -- 1 , 2 , . . . , ( M - 1 ) , and those coefficients of fo and fM that are arbitrarily assignable, one must have
Theorem
,n _> n -
z § {i(.-
,-)/M}
M-
~
s _.~.1
Proof: C o n s i d e r i n g fo to b e m o n i c , it follows f r o m t h e P r o p e r t i e s 2 t h r o u g h 5 o b s e r v e d a b o v e t h a t t h e t o t a l n u m b e r of a r b i t r a r y coefficients i n v o l v e d in e q n . ( 2 9 ) is M-1
nx = m +
~[rn+n+l--Z
+{i(n-r)/M}]+(rn+l)
i=l
B u t t h e t o t a l n u m b e r of i n d e p e n d e n t l y v a r i a b l e p a r a m e t r s in t h e c o n t r o l l e r given by e q n . ( 5 ) is n2 -- (2rn + 1 ) M . Clearly, for t h e coefficients of t h e C E to be a r b i t r a r i l y a s s i g n a b l e o n e m u s t h a v e n2 > n l , w h i c h l e a d s to eqn.(31). !-!
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
! C(z) !
149
] /___~
,G(z) [ (2N)
F i g u r e 3: S c h e m e for zero p l a c e m e n t of even to even t r a n s f e r f u n c t i o n s
To o b t a i n t h e p a r a m e t e r values of t h e r n t h o r d e r (rn o b t a i n e d as a b o v e ) M - p e r i o d i c c o n t r o l l e r to achieve a d e s i r e d C E , o n e h a s t o e q u a t e t h e coefficients of t h e different p o w e r s of z M in .f~(zM), i = 0, 1, . . . , M , t o t h e d e s i r e d v a l u e s l e a d i n g to a set of n o n - l i n e a r s i m u l t a n e o u s e q u a t i o n s w h i c h is t o be s o l v e d t o o b t a i n t h e v a l u e s for c~k a n d dik. W h a t should, however, be the desired CE when robust G M c o m p e n s a tion is s o u g h t ? T o a n s w e r this q u e s t i o n it w o u l d b e h e l p f u l first t o e x t e n d t h e c o n c e p t of zeros a n d e x a m i n e h o w s u c h zeros c a n b e r e l o c a t e d u s i n g periodic controllers.
V A
Zero Placement Conventional
by Periodic
Controllers
Approach
C o n v e n t i o n a l l y zeros are u n d e r s t o o d t o be t h e r o o t s of t h e n u m e r a t o r p o l y n o m i a l of a given t r a n s f e r f u n c t i o n . Zero p l a c e m e n t in t h i s s e n s e h a s b e e n d o n e in [7] e m p l o y i n g t h e following a p p r o a c h . T h e set u p c o n s i s t s of a L D T I c o m p e n s a t o r C(z) p u t in series w i t h t h e given p l a n t G(z) a n d t w o 2 T - s a m p l e r s , o n e at t h e i n p u t a n d t h e o t h e r at t h e o u t p u t of t h e GC p a i r . (See F i g . 3 . ) T h u s w h i l e C e x c i t e s G at T i n t e r v a l s of t i m e , t h e p a i r GC as a w h o l e r e c e i v e s i n p u t a n d yields o u t p u t o n l y at 2 T i n t e r v a l s of t i m e . N o w , v i e w i n g G(z) as a 2 - p e r i o d i c p l a n t its e q u i v a l e n t 2 x 2 t r a n s f e r m a t r i x c a n be o b t a i n e d following C o r o l l a r y 1.1 as
G(=2) =
[G,(~ ~) ~-2G2(~2) ] G2(~~) G,(~2)
w h e r e G1 a n d G2 s a t i s f y
G(~) = G,(~ ~) + ~-'V~(~ ~) S i m i l a r l y , C(z) m a y be a s s u m e d to h a v e t h e 2 x 2 f o r m
c(~) =
(~) ~-2C2(z2) ] IL cl Ct(z2) C2(z2) r
where
c(~) = c~(~ =) + ~-~c~(~ ~)
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SARIT K. DAS AND R K. RAJAGOPALAN
T h e n t h e t r a n s f e r f u n c t i o n b e t w e e n t h e even i n s t a n t i n p u t to C a n d t h e even i n s t a n t o u t p u t of G b e c o m e s
T~,(~ :) = G~ ( ~ ) c ~ (~=) + ~-2G2(~2)C:(~) It is clear t h a t by c h o o s i n g C1 a n d C~ s u i t a b l y t h e n u m e r a t o r of Tee can be set a r b i t r a r i l y , t h u s effecting t h e zero p l a c e m e n t of t h e o v e r a l l s y s t e m . A s i m p l e way to c h o o s e C1 a n d C2 is to a s s u m e
C~(z 2) -- (C~,mz 2m + C;,,~_xz 2(m-1) + . . .
+ C~,o)/z 2'~,
i = 1, 2
For rn > ( n 1), w h e r e n is t h e n u m b e r of p o l e s of G ( z ) , it w o u l d be p o s s i b l e to p l a c e all t h e finite zeros of Tee a r b i t r a r i l y . E x a m p l e 1 S u p p o s e it is r e q u i r e d to shift t h e zeros of t h e p l a n t G ( z ) -( z - 2 ) / ( z - 3) to origin. To this end first find G1 a n d G2 as G , = (z 2 -
6)/(z 2-9)
a n d G2 = z 2 1 ( z 2 - 9 )
N e x t c h o o s e t h e c o n t r o l l e r as
c ( ~ ) = C~o + C.o/~ We
require c,0(~ ~ - 6)/(z 2 _ 9) + C 2 o l ( Z 2 -- 9) -- z 2 1 ( z 2 -- 9)
T h i s yields Ct0 = 1 a n d C20 = 6. It m a y be e m p h a s i z e d t h a t , in reality, t h e a b o v e p r o c e d u r e p l a c e s arb i t r a r i l y t h e zeros of t h e e v e n to e v e n t r a n s f e r f u n c t i o n only. O n e m a y , of course, use t h e s a m e t e c h n i q u e to p l a c e t h e zeros of t h e e v e n to o d d , o d d to e v e n , or t h e o d d to o d d t r a n s f e r f u n c t i o n i n s t e a d .
B
Characteristic
Equation
Approach
B a s e d on t h e u n d e r s t a n d i n g t h a t t h e closed loop r o o t loci e n d u p at t h e o p e n loop zeros, t h e zeros of a s y s t e m m a y c o n v e n i e n t l y b e d e f i n e d to be t h e roots of the closed loop c h a r a c t e r i s t i c e q u a t i o n with the loop gain tending to oo. A l t h o u g h for S I S O L D T I s y s t e m s t h i s a n d t h e c o n v e n t i o n a l d e f i n i t i o n of zeros are c o m p l e t e l y e q u i v a l e n t , for p e r i o d i c s y s t e m s t h e f o r m e r is m o r e a d v a n t a g e o u s . T h i s is b e c a u s e , as n o t e d earlier, t h e i n p u t - o u t p u t d e s c r i p t i o n of a p e r i o d i c s y s t e m is o b t a i n e d e i t h e r in t h e f o r m of a p e r i o d i c coefficient t r a n s f e r f u n c t i o n or as a M I M O L D T I t r a n s f e r m a t r i x , in n e i t h e r of w h i c h t h e c o n c e p t of n u m e r a t o r r o o t s is r e a d i l y a p p l i c a b l e . T h e c o n c e p t of closed loop C E is, on t h e o t h e r h a n d , u n i v e r s a l .
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
151
R e c a l l t h a t t h e C E of a M - p e r i o d i c s y s t e m is of t h e f o r m of e q n . ( 2 9 ) . If t h e a l t e r n a t i v e d e f i n i t i o n of zeros is now e m p l o y e d t h e n t h e coefficient of t h e h i g h e s t p o w e r e d K - t e r m , i.e. f M , w o u l d s t a n d for t h e z e r o - p o l y n o m i a l of t h e s y s t e m . H o w e v e r , n o t e f r o m t h e P r o p e r t y 4 o b s e r v e d in t h e p r e c e d i n g s e c t i o n t h a t t h e zeros of fM are n o t all a r b i t r a r y . H e n c e for a r b i t r a r y zero p l a c e m e n t o n e m u s t m a k e f M =-- O. If t h i s is d o n e t h e n f M - 1 w o u l d b e c o m e t h e z e r o - p o l y n o m i a l of t h e s y s t e m . F o r M -- 2, this l e a d s to a m e a n i n g f u l r e s u l t since t h e n t h e C E t a k e s t h e f o r m
f o ( z 2) + g f l (z 2) = 0 a n d it is easy to i d e n t i f y f l as t h e z e r o - p o l y n o m i a l . ( N o t e t h a t in [5] zero p l a c e m e n t u s i n g 2 - p e r i o d i c c o n t r o l l e r s b a s e d on t h e a b o v e i d e a h a s b e e n p e r f o r m e d . N o t e also f r o m e q n . ( 2 0 ) t h a t t h e f2 t e r m for s u c h a s y s t e m c a n be m a d e zero by c h o o s i n g ql (z) as any o n e of t h e four p o l y n o m i a l s q l ( z ) ----=t=qo(=t=z).) For M > 2, h o w e v e r , it is n o t e n t i r e l y s a t i s f a c t o r y to define f M - ~ as t h e z e r o - p o l y n o m i a l while t h e t e r m s K i f i , i = 1 , 2 , . . . , ( M 2), are also p r e s e n t in t h e C E . T h i s is b e c a u s e t h e r o o t loci t h e n d e p e n d n o t o n l y on t h e zeros of f M - 1 b u t on t h o s e of f l , - . . , f M - 2 as well. T h u s t h e r e is a n e e d for a s t r i c t e r d e f i n i t i o n of t h e zeros.
Definition 1 I f the CE of a system with loop gain K is of the f o r m fo(Z + 1 ) + K i f , ( z ~ ) = 0 t h e n ~he zeros of fi will be called the stric~ zeros of type-i of the system.
T h u s to p l a c e t h e zeros of a M - p e r i o d i c s y s t e m in t h e s t r i c t s e n s e not o n l y f M b u t also fk, k = 1 , 2 , . . . , ( M 1), k r i, h a v e to b e m a d e i d e n t i c a l l y e q u a l to zero. N e x t we h a v e t h e following r e s u l t r e g a r d i n g t h e s t r i c t zero p l a c e m e n t c a p a b i l i t y of M - p e r i o d i c c o n t r o l l e r s . For c o n v e n i e n c e , h o w e v e r , t h e t e r m ' s t r i c t ' will h e n c e f o r t h be d r o p p e d . Theorem 5 Consider a system that has a n-pole r-zero plant along with a M - p e r i o d i c rnth order controller such that it leads to a C E of the f o r m of eqn.(29). A s s u m e that CondiZion 1 is satisfied. Then the following are true:
1. Arbitrary zero p l a c e m e n t of type-i is possible if 0 < i < M M > 1, and rn satisfies eqn.(31).
with
~. For given values of M and i lying in the ranges noted above the n u m b e r o f r e l o c a t a b l e p l a n t zeros is r 1 -- . -
Z + {i(n-
r)/M}
(32)
152
SARIT K. DAS AND E K. RAJAGOPALAN
3. For zero p l a c e m e n t it is m o s t a d v a n t a g e o u s to have i = 1. Proof: 1. F r o m t h e definition of zeros it is o b v i o u s t h a t i s h o u l d b e g r e a t e r t h a n zero. A g a i n since t h e zeros of f M are n o t all a r b i t r a r y , i m u s t be less t h a n M a n d M g r e a t e r t h a n unity. F u r t h e r it follows f r o m T h e o r e m 4 t h a t rn s h o u l d satisfy e q n . ( 3 1 ) . It follows f r o m e q n . ( 3 0 ) t h a t t h e n u m b e r of zeros of fl is ~rt-brl, w h e r e r l is as given by e q n . ( 3 2 ) . Now since t h e c o n t r o l l e r is a r u t h o r d e r one, m of t h e rn + r l zeros m a y be c o n s i d e r e d to be b e l o n g i n g to t h e c o n t r o l l e r , a n d t h e r e m a i n i n g r l can be visuslized to be t h e r e l o c a t e d p l a n t zeros. 3. It follows f r o m e q n . ( 3 2 ) t h a t for a given M , rl is m a x i m u m w h e n i = 1. A g a i n , it is also e v i d e n t from e q n . ( 3 2 ) t h a t t h e v a l u e of M r e q u i r e d t o m a x i m i z e r l will be m i n i m u m w h e n i -- 1. H e n c e it is m o s t a d v a n t a g e o u s to h a v e i -- 1. El F u r t h e r , t h e following c o r o l l a r y gives t h e m a x i m u m n u m b e r of zeros t h a t are r e l o c a t a b l e for a given n - p o l e r - z e r o p l a n t .
Corollary
5.1
1. For bicausal (i.e. n = r ) plants all the p l a n t zeros can be relocated using a 2-periodic controller. ~. For strictly proper (i.e. n > r ) plants, the n u m b e r ( r l ) o f relocatable p l a n t zeros lies within the limits r <_ rl <_ ( n - 1), the u p p e r limit being obtained when M ~_ i ( n - r). Proof: Follows easily f r o m e q n . ( 3 2 ) .
D
R e m a r k 5 . 2 : C o r o l l a r y 5.1 signifies t h a t p e r i o d i c c o n t r o l l e r s (i) can a l w a y s r e l o c a t e t h e given r finite p l a n t zeros, a n d (ii) can, m o r e o v e r , r e l o c a t e all but one of t h e p l a n t zeros t h a t were o r i g i n a l l y at infinity. 1-] It m a y b e n o t e d in this c o n t e x t f r o m e q n s . ( 2 1 ) a n d (22) t h a t , for M = 2, t h e C E c o r r e s p o n d i n g to an e v e n - o u t p u t or an e v e n - i n p u t c o n t r o l l e r c o n t a i n s only K ~ a n d K 1 t e r m s , a n d h e n c e l e a d s a u t o m a t i c a l l y to t y p e - 1 zero p l a c e m e n t . T h i s r e s u l t is t r u e as well for g e n e r a l M - p e r i o d i c c o n t r o l l e r s h a v i n g e i t h e r an i n p u t or an o u t p u t M T s a m p l e r . T h i s can be r e a d i l y seen by n o t i n g t h a t t h e e q u i v a l e n t M x M t r a n s f e r m a t r i x for such a c o n t r o l l e r will h a v e e i t h e r o n l y o n e n o n - z e r o c o l u m n or o n l y o n e n o n - z e r o row, a n d t h e c o r r e s p o n d i n g d e t ( I + K ( ~ C ) w o u l d h a v e only K ~ a n d K 1 t e r m s . In t h e n e x t s e c t i o n we s h o w how t h e zero p l a c e m e n t c a p a b i l i t y of periodic c o n t r o l l e r s c a n b e e m p l o y e d to p r o v i d e i m p r o v e d G M c o m p e n s a t i o n to L D T I p l a n t s .
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
VI
Gain Margin Compensation odic Controllers
153
u s i n g Peri-
For a closed l o o p s y s t e m h a v i n g a loop g a i n K , t h e t e r m G M refers to t h e r a t i o Kmax/Kmin w h e r e K,~== a n d Kmln are, r e s p e c t i v e l y , t h e m a x i m u m a n d t h e m i n i m u m v a l u e s of a r a n g e of K for which t h e s y s t e m is s t a b l e . Since t h e b r a n c h e s of t h e r o o t loci of a closed l o o p s y s t e m s t a r t f r o m t h e o p e n l o o p p o l e s a n d e n d at t h e zeros, it follows t h a t t h e m a x i m u m G M t h a t a L D T I c o m p e n s a t o r w o u l d b e able to a c h i e v e for a p l a n t t h a t h a s u n s t a b l e p o l e ( s ) as well as N M P zero(s) is a finite n u m b e r . (In A p p e n d i x C we p r e s e n t briefly t h e m e t h o d s of [9] a n d [10] for b e s t G M c o m p e n s a t i o n using L D T I c o m p e n s a t o r s . ) We now show t h a t p e r i o d i c c o n t r o l l e r s c a n do b e t t e r in t h i s r e s p e c t .
A
Bicausal
Plants
It follows f r o m c o r o l l a r y 5.1 t h a t it is p o s s i b l e to p l a c e all t h e zeros o f such a p l a n t w i t h i n t h e u n i t circle u s i n g a 2 - p e r i o d i c c o n t r o l l e r . T h i s w o u l d o b v i o u s l y lead to infinite G M c o m p e n s a t i o n , w h e r e v e r be t h e poles. T h e zero p l a c e m e n t can be c a r r i e d o u t using e i t h e r t h e c o n v e n t i o n a l a p p r o a c h of [7] or t h e C E a p p r o a c h of [5]. In E x a m p l e 1 t h e f o r m e r a p p r o a c h was u s e d . In t h e following e x a m p l e we use t h e l a t t e r . Example 2 Given the plant ( z - 2)/(z3), we w o u l d like to h a v e a 2p e r i o d i c c o n t r o l l e r of t h e f o r m of e q n . ( 5 ) such t h a t t h e s y s t e m zeros are p l a c e d at origin. Since n -- r -- 1, we h a v e f r o m e q n . ( 3 1 ) t h e m i n i m u m v a l u e of t h e c o n t r o l l e r o r d e r r e q u i r e d as rn --- 1. H e n c e t h e c o n t r o l l e r p a r a m e t e r s t o be f o u n d are p0(z)
=
z+c0o
pl(z)
=
c01
qo(z)
=
dtoz + doo
ql(z)
=
dttz+d0t
Let q l ( z ) - qo(z), i.e. dot --- d00 a n d dtz ---- dt0. T h i s m a k e s t h e c o n t r o l l e r an e v e n - o u t p u t o n e w i t h t h e C E as given by e q n . ( 2 1 ) . Now, t h e l o c a t i o n of t h e c o n t r o l l e r p o l e s c a n be a r b i t r a r y in t h i s p r o b l e m a n d we c h o o s e to p l a c e t h a s a m e at origin. F r o m e q n . ( 2 1 ) we t h e n h a v e no(~)po(-~)
- pl(~)p~(-~)
= -[(~
2
+ ,=oo)(~ - ~oo) + %1] = -
Z2
(33)
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SARIT K. DAS AND R K. RAJAGOPALAN
Also since all t h e zeros a s s o c i a t e d w i t h t h e K - t e r m origin we h a v e h(z) + h(-,)
are t o be p l a c e d at
= 2~ 9
h ( z ) = (z -- 3)(z + 2 ) { ( z + coo)(dloz - doo) + c o l ( d l l z + d01)} (34) T h e s i m u l t a n e o u s e q n s . ( 3 3 ) a n d (34) are n o n - l i n e a r in n a t u r e a n d h e n c e h a v e m a n y s o l u t i o n s . O n e s o l u t i o n is dxl = dl0 ---- 1, d01 = d00 -- 6, Col -" Coo "- 0. ( N o t e t h a t t h e c o n t r o l l e r t h u s o b t a i n e d m a y also be r e a l i z e d as a L D T I t r a n s f e r f u n c t i o n C ( z ) = 1 + 6 / z followed by an e v e n i n s t a n t s a m p l e r . ) T h e e n s u i n g C E b e c o m e s z2[z 2 - 9 - 2 K z 2] = 0, w h i c h s h o w s t h a t t h e s y s t e m will be s t a b l e for K > 5 a n d K < - 4 . D where
In t h e a b o v e e x a m p l e t h e c o n t r o l l e r c o u l d also h a v e b e e n c h o s e n to b e an e v e n - i n p u t one. T h i s w o u l d h a v e y i e l d e d t h e c o n t r o l l e r as a C'(z) -- 1 -b 6 / z p r e c e d e d by an e v e n - i n s t a n t s a m p l e r . Now n o t e f r o m E x a m p l e 1 t h a t t h e o p e n - l o o p c o n v e n t i o n a l zero p l a c e m e n t a p p r o a c h leads to t h e s a m e cont r o l l e r for t h e s a m e p l a n t . T h i s s h o w s t h a t so far as G M c o m p e n s a t i o n of b i c a u s a l p l a n t s is c o n c e r n e d t h e t w o zero p l a c e m e n t a p p r o a c h e s a r e e q u i v a lent. It w o u l d , h o w e v e r , be seen in t h e n e x t s e c t i o n t h a t for s t r i c t l y p r o p e r p l a n t s t h e C E a p p r o a c h is m o r e a d v a n t a g e o u s . T h i s is b e c a u s e for t h e G M c o m p e n s a t i o n of s u c h p l a n t s t h e l o c a t i o n of t h e c o n t r o l l e r p o l e s is also of i m p o r t a n c e . U s i n g t h e C E a p p r o a c h t h e c o n t r o l l e r pole p l a c e m e n t a n d t h e s y s t e m zero p l a c e m e n t c a n b o t h b e d o n e in a single s t e p . Using t h e conv e n t i o n a ] a p p r o a c h , on t h e o t h e r h a n d , first zero p l a c e m e n t is to b e d o n e in o p e n loop, a n d t h e n t h e loop is to b e closed p l a c i n g an a d d i t i o n a l t r a n s f e r f u n c t i o n in t h e f e e d b a c k p a t h so t h a t t h e r e q u i r e d l o o p t r a n s f e r f u n c t i o n is achieved. B
Strictly
Proper
Plants
F o r s t r i c t l y p r o p e r p l a n t s we h a v e f r o m C o r o l l a r y 5.1 t h a t e v e n w i t h p e r i o d i c c o n t r o l l e r s t h e r e w o u l d c o n t i n u e to be at least o n e zero at infinity. T h i s p r e c l u d e s t h e p o s s i b i l i t y of infinte G M c o m p e n s a t i o n of such ( u n s t a b l e ) p l a n t s . N e v e r t h e l e s s it is r e a s o n a b l e to e x p e c t t h a t by p l a c i n g t h e zeros suitably periodic controllers may achieve b e t t e r G M t h a n o b t a i n a b l e using L D T I c o n t r o l l e r s . W e p r e s e n t a n u m e r i c a l e x a m p l e to s h o w h o w p e r i o d i c c o n t r o l l e r s m a y a c h i e v e s u c h i m p r o v e m e n t in G M . Example 3 S u p p o s e t h a t it is r e q u i r e d to m a x i m i z e t h e G M of t h e p l a n t G(z) = (z- 3 ) / ( z - 2 ) ( z - 5/3) using a 2-periodic controller following the zero p l a c e m e n t a p p r o a c h . It is o b v i o u s t h a t for b e s t G M c o m p e n s a t i o n t h e zero s h o u l d b e r e l o c a t e d inside t h e u n i t circle, say, at t h e origin. T h a t is, we w a n t t h e C E of t h e
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
155
2 - p e r i o d i c a l l y c o m p e n s a t e d s y s t e m to be of t h e f o r m (2
_ 4)(~2 _ ~5/9)f~(~2)
+ K~2f~(2)
= 0
w h e r e f t a n d f2 are n o w to be c h o s e n so as t o m a x i m i z e t h e G M . N o t e , h o w e v e r , t h a t t h i s p r o b l e m m a y e q u i v a l e n t l y be v i s u a l i z e d to b e o n e of finding t h e b e s t G M c o m p e n s a t i o n of t h e p l a n t z2/(z 2 - 4 ) ( z 2 - 2 5 / 9 ) u s i n g a L D T I c o m p e n s a t o r f2(z2)/ft(z2). Now it follows f r o m [10] (see A p p e n d i x C) t h a t t h e o p t i m u m v a l u e s of f t a n d f2 are f~(~)
= (~
f 2 ( ~ ~) = ( ~ -
-
1/4)(~ ~ 1.79)(~ ~ -
9/25) 0.56)
t h e c o r r e s p o n d i n g G M b e i n g 1.43. ( N o t e t h a t t h e m a x i m u m G M o b t a i n a b l e for t h e o r i g i n a l p l a n t u s i n g L D T I c o m p e n s a t i o n is o n l y 1.116.) To a c h i e v e t h e a b o v e C E a s e c o n d o r d e r c o n t r o l l e r w o u l d suffice a n d it h a s b e e n s h o w n in [5] t h a t o n e s u c h c o n t r o l l e r is [d20 -- 1, d2t "" - 1 , dr0 -" d t t -- - 1 . 6 9 , d00 "- dot -- 0, ct0 = 5.47, ctt -- 4.11, coo -- 6.207 a n d cot = - - 6 . 2 ] . It m a y b e n o t e d in t h i s c o n t e x t t h a t t h e s a m e G M c a n also b e a c h i e v e d u s i n g t h e c o n v e n t i o n a l zero p l a c e m e n t a p p r o a c h of [7]. F o l l o w i n g S e c t i o n V - A it is e a s y to s h o w t h a t c h o o s i n g a L D T I c o n t r o l l e r g i v e n by C'(z) = (4.5z~+78.034z-60.126)/z 2 o n e c a n m a k e t h e e v e n t o e v e n t r a n s f e r f u n c t i o n of t h e G C p a i r to b e (z 2 - 1.79)(z 2 - 0 . 5 6 ) / z 2 ( z 2 - 4 ) ( z 2 - 2 5 / 9 ) . N o w if t h e loop is closed t h r o u g h a t r a n s f e r f u n c t i o n L = z4/(z 2 - 1 / 4 ) ( z 2 - 9 / 2 5 ) , t h e n we w o u l d o b v i o u s l y h a v e t h e s a m e C E as c o r r e s p o n d s t o t h e m a x i m u m G M . N o t e , h o w e v e r , t h a t t h e o r d e r of t h e c o n t r o l l e r r e q u i r e d in t h i s a p p r o a c h is m u c h h i g h e r t h a n t h e s a m e r e q u i r e d in t h e C E a p p r o a c h . [-1 It is n o w n a t u r a l t o ask if t h e G M of t h e a b o v e p l a n t c a n b e i m p r o v e d f u r t h e r u s i n g c o n t r o l l e r s w i t h p e r i o d i c i t y M > 2. In fact, in g e n e r a l o n e w o u l d like to k n o w h o w a given p l a n t is to be o p t i m a l l y c o m p e n s a t e d u s i n g periodic controllers.
1
O p t i m u m G M C o m p e n s a t i o n using Zero P l a c e m e n t
Since the root loci end at the zeros, it appears logical to infer that for better G M compensation the n u m b e r of N M P zeros should be minimized. Now, we have from Corollary 5.1 that for a n-pole, r-zero (n > r) plant periodic controllers can relocate upto ( n - 1) zeros. T h u s it appears at first sight that the best G M will be obtained w h e n there is only one zero at infinity, for which one requires the periodicity to be M >_ i(n - r), i being the type of the relocated zeros. O n the other hand, note from Property 3 given in Section IV that a M-periodic controller causes the plant poles
156
SARIT K. DAS AND P. K. RAJAGOPALAN
to be t h e M t h p o w e r s of t h e i r original values, t h u s p u s h i n g t h e u n s t a b l e poles f u r t h e r a w a y f r o m t h e u n i t circle a n d , c o n s e q u e n t l y , w o r s e n i n g t h e G M p r o s p e c t s . So we h a v e t h a t an i n c r e a s e in t h e v a l u e of M h a s b o t h a p o s i t i v e a n d a n e g a t i v e effect, a n d it is n o t clear (i) if p e r i o d i c c o n t r o l l e r s can i m p r o v e t h e G M of all s t r i c t l y p r o p e r p l a n t s , a n d (ii) how, in g e n e r a l , to find t h e o p t i m u m M t h a t yields t h e b e s t G M for a given p l a n t . In t h i s s e c t i o n we a n s w e r t h e s e q u e s t i o n s . F i r s t , h o w e v e r , we find o u t t h e t y p e of zero p l a c e m e n t t h a t s h o u l d be effected for b e s t results. L e m m a 2 For o p t i m u m G M c o m p e n s a t i o n using the zero p l a c e m e n t approach type-1 zeros need only be considered.
Proof: We h a v e f r o m e q n . ( 3 2 ) t h a t for a given M t h e r e will be ( n - r l ) = Z + {i(n-r)/M} t y p e - / u n c o n t r o l l a b l e zeros at infinity. O b v i o u s l y t h e n u m b e r of such zeros will be m i n i m u m w h e n i = 1, a n d , o t h e r f a c t o r s r e m a i n i n g equal, t h i s a l o n e is sufficient to p r o v e t h e result. In reality, m o r e o v e r , a choice of i > 1 m a k e s t h e G M p r o s p e c t s still worse since, as follows f r o m D e f i n i t i o n 1, t h e C E t h e n h a s t h e gain t e r m not m e r e l y as K b u t K i. D H e n c e f o r t h t h e t e r m ' z e r o ' will only m e a n a t y p e - 1 zero. We now p r e s e n t o u r m a i n r e s u l t . Theorem
6
W i t h regard to G M c o m p e n s a t i o n using zero p l a c e m e n t of a
.~i~tty p~op~. ~-pot~. ~-~.o /or. ( . , . ) 4 pz~.t 1. if all the r finite zeros are m i n i m a m phase then M -- 1 ( i . e i n v a r i a n t controller) yields the best GM, 2. in general, the o p t i m u m M is a p r i m e n u m b e r <_ ( n f o r n - r = 1, M = 2 is also to be considered.
the t i m e
r). H o w e v e r ,
Proof: C o n s i d e r two c o n t r o l l e r s , o n e w i t h p e r i o d i c i t y M1 a n d t h e o t h e r w i t h M2, a n d s u p p o s e t h a t in each case all t h e r e l o c a t a b l e p l a n t zeros h a v e b e e n p l a c e d at origin so t h a t t h e b e s t G M can be o b t a i n e d w i t h t h a t p e r i o d i c i t y . T h e n for e a c h M i , i = 1, 2, t h e s y s t e m C E t a k e s (following P r o p e r t i e s 3 a n d 4 n o t e d in S e c t i o n IV) t h e f o r m
w h e r e ak are t h e o r i g i n a l p l a n t poles, foi a n d f l l are t h e r u t h d e g r e e p o l y n o m i a l s i n t r o d u c e d by t h e c o n t r o l l e r to p r o v i d e G M c o m p e n s a t i o n , a n d ri is t h e n u m b e r of r e l o c a t e d p l a n t zeros which, following e q n . ( 3 2 ) , equaJs
ri = n - Z + {(n - r ) / M i }
(36)
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
157
Clearly, f r o m e q n . ( 3 5 ) t h e o r i g i n a l p a i n t c a n b e v i s u a l i z e d t o h a v e b e c o m e
{z~'}"/fi(~'
G~(~') =
- ~k~'),
(3T)
i = 1,2
k---1
Let M2 = IM1 w h e r e 1 is an i n t e g e r . T h e n , d e f i n i n g Zl "-- z M1
and
z2 :-- z M2
(38)
one c a n r e w r i t e G2 in t e r m s of G1 as n
G~(~)
=
z;,/Hcz,-
of,)
k-----I
rl ( l r 2 - r , )
--
z1 z1
/f(zl)
f_ _i ( z l
M1 )
-- a k
k----1
(39) w h e r e f ( z ~ ) , it suffices to n o t e , is a p o l y n o m i a l of d e g r e e n ( l we h a v e f r o m e q n . ( 3 6 ) t h a t lr2 - r l
=
.(z-
<
n ( l - - 1)
~) - [ t z + { ( ~
-
,)llM~}-
Z+{(~ -
1). F u r t h e r
,)IM~}] (40)
(/r2--rl)
T h i s m e a n s t h a t z, / . f ( z l ) is a p r o p e r or s t r i c t l y p r o p e r t r a n s f e r function. It is t h e n o b v i o u s f r o m e q n . ( 3 9 ) t h a t t h e G M a t t a i n a b l e for G2 c a n n o t b e b e t t e r t h a n t h a t a t t a i n a b l e for G1. Since M1 a n d I are a r b i t r a r y , t h e following c o n c l u s i o n s c a n i m m e d i a t e l y be d r a w n 1. If G1 c o r r e s p o n d i n g to M1 = 1 (i.e. t h e g i v e n p l a n t itself) is of t h e f o r m of e q n . ( 3 7 ) (i.e. h a s no finite zero o u t s i d e t h e u n i t circle), t h e n t h e M = 1 (i.e. t h e t i m e i n v a r i a n t ) c o n t r o l l e r yields t h e b e s t r e s u l t s . ( T h i s is so b e c a u s e a n y o t h e r value of M c a n o n l y be a n i n t e g r a l m u l t i p l e of u n i t y . ) 2. Else, h i g h e r v a l u e s of M , considered.
b u t o n l y p r i m e n u m b e r s a r e also t o be
It n o w r e m a i n s to be s h o w n t h a t M > ( n - r) n e e d n o t b e c o n s i d e r e d . To t h i s e n d n o t e f r o m c o r o l l a r y 5.1 t h a t for all M >_ ( n - r) t h e s y s t e m w o u l d h a v e o n e zero at infinity. C o n s i d e r s u c h a M a n d a s s u m e t h a t t h e
158
SARIT K. DAS AND R K. RAJAGOPALAN
( n - - 1 ) re lo c at able zeros h a v e b e e n p l a c e d at origin, l e a d i n g to t h e e q u i v a l e n t plant configuration n
G1 = z 1n - - 1 / I ~ ( z l - - a ~ )
w h e r e zl "= z M
k-----1
F o l l o w i n g [9] (see A p p e n d i x C), it is n o t difficult to see t h a t t h e m a x i m u m n G M a t t a i n a b l e for t h i s p l a n t is (1 + r/)2/(1 - r/) 2 w h e r e r/ -- 1/1-Ikffil a ~ . Clearly, ~/ will be s m a l l e r for l a r g e r M , m a k i n g t h e G M lower. H e n c e M > ( n - r) n e e d n o t be c o n s i d e r e d . F i n a l l y , since t i m e i n v a r i a n t c o n t r o l l e r s do n o t h a v e t h e zero p l a c e m e n t c a p a b i l i t y , it is o b v i o u s t h a t for t h e case of n - r -- 1, M -- 2 is also to be considered. U] W e are n o w in a p o s i t i o n to o b t a i n t h e b e s t G M c o m p e n s a t i o n for a given p l a n t v i a t h e zero p l a c e m e n t a p p r o a c h . T h e s t e p s to be followed for this p u r p o s e are: 1. F o l l w i n g T h e o r e m 6 i d e n t i f y t h e values of M t h a t a r e to be c o n s i d e r e d . 2. F o r each of t h e s e v a l u e s of M o b t a i n t h e e q u i v a l e n t p l a n t [as given by eqn.(37)] a s s u m i n g t h a t all t h e r e l o c a t a b l e zeros h a v e b e e n p l a c e d at origin. 3. F o r e a c h of t h e s e e q u i v a l e n t p l a n t s o b t a i n G M M ( w h i c h s t a n d s for t h e b e s t G M o b t a i n a b l e u s i n g M - p e r i o d i c c o n t r o l l e r ) u s i n g t h e app r o a c h of [9] or [10]. T h e n o b t a i n t h e m a x i m u m G M as G M m a ~ -m a x [ G M M ] , M v a r y i n g over t h e r a n g e given by s t e p 1 a b o v e . M
4. F o r M = M m a x c o r r e s p o n d i n g to G M , n a x , o b t a i n t h e c o m p e n s a t o r a n d t h e c o m p l e t e C E u s i n g t h e a p p r o a c h of [10]. 5. F i n a l l y , to o b t a i n t h e p a r a m e t e r s of t h e p e r i o d i c g a i n c o n t r o l l e r t h a t a c h i e v e s t h e a b o v e C E , first check if t h e c o n t r o l l e r o r d e r rn as o b t a i n e d in s t e p 4 a b o v e satisfies t h e lower l i m i t c o n d i t i o n given by T h e o r e m 4. If it is satisfied t h e n this rn = rnmin, t h e m i n i m u m o r d e r of t h e c o n t r o l l e r r e q u i r e d . O t h e r w i s e , rnmir, will be t h e s m a l l e s t v a l u e of rn t h a t satisfies e q n . ( 3 1 ) . N e x t e q u a t e t h e coefficients of t h e like p o w e r s of t h e K - t e r m s of e q n . ( 2 8 ) to t h o s e of t h e d e s i r e d C E to g e t a set of n o n - l i n e a r e q u a t i o n s s o l v i n g which t h e c o n t r o l l e r p a r a m e t e r s c a n be obtained.
We n o w p r e s e n t a few e x a m p l e s of o p t i m u m G M c o m p e n s a t i o n of s t r i c t l y proper plants.
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
Example
4 C o n s i d e r t h e (2, 1 ) - p l a n t ( z -
159
1.5), Ibl > 1.
b)/z(z-
i) V a l u e s of M t o b e c o n s i d e r e d a r e 1 a n d 2. ii) F o r M = 2, t h e e q u i v a l e n t p l a n t w o u l d , o n z e r o r e l o c a t i o n , b e 1 / ( z t 2.25), for w h i c h G M M zt(zt
--- 6.76 a n d t h e c o r r e s p o n d i n g
C E is
-- 2 . 2 5 ) ( z t -- 1 / 2 . 2 5 ) + a K z 2 t = 0
C l e a r l y m = r n m i n = 1. N o w , o n e s e t o f t h e p a r a m e t e r
v a l u e s o f t h e first
o r d e r c o n t r o l l e r t h a t a c h i e v e s t h i s is o b t a i n e d t o b e qo(z)
-- z + 1.2b/(b
+
1.2)
qt ( z ) - - q o ( - - z ) p o ( z ) -- z + 1 / 1 . 5
pl(~) = o and the corresponding
c~ b e c o m e s
= 2(1.5 - b)(b + o . 6 6 7 ) / ( b + 1.2) iii) F o l l o w i n g [9] it is s e e n t h a t , for M
=
1, G M M
> 6.76 if a n d o n l y if
-r < b < 1.083. T h u s for b > 1.083 2 - p e r i o d i c c o n t r o l l e r s y i e l d b e t t e r results than time invariant ones. Example
5 C o n s i d e r t h e (3, 1 ) - p l a n t ( z -
b)/z(z
- 1.5)(z-
2).
i) T h e v a l u e s o f M t o b e c o n s i d e r e d a r e 1 a n d 2. ii) F o r M "- 2, t h e e q u i v a l e n t p l a n t a f t e r t w o z e r o s a r e r e l o c a t e d t o o r i g i n would be zt/(zt - 2.25)(zt -4), for w h i c h G M M -- 1.56 a n d t h e c o r r e s p o n d i n g C E is z t ( z t -- 2 . 2 5 ) ( Z l -- 4 ) ( z t -- 1 / 2 . 2 5 ) ( Z l -- 1 / 4 ) + o ~ K z ~ ( z t H e r e rn -- rn,,+in = 2. C E for b - 1.5 is
- 1 . 6 ) ( Z l - 1 / 1 . 6 ) -- 0
Now a second order controller which achieves this
q o ( z ) = 0 . 3 3 6 z 2 + 0 . 1 6 9 z + 0.946 qt ( z ) -- q o ( z ) p o ( z ) -- z 2 + 3 . 6 8 2 z + 1.109 P l ( z ) -- - - 3 . 2 6 8 z + 1.05 and the corresponding
c~ is a = 0.799.
iii) F o r M = 1, G M M
> 1.56 if a n d o n l y i f - 1 . 2 6 5
< b < 1.25.
T h u s for
values of b outside this range the 2-periodic controller yields better results.
160
SARIT K. DAS AND E K. RAJAGOPALAN
Example
6
C o n s i d e r t h e (4, 1 ) - p l a n t ( z -
b)/z2(z-
1.5)(z-
2).
i) M = 1, 2 a n d 3 n e e d o n l y be c o n s i d e r e d . ii) F o r M ---- 2, o n l y two z e r o s c a n be r e l o c a t e d to origin. T h e n t h e e q u i v a lent p l a n t w o u l d be 1 / ( Z l - 2 . 2 5 ) ( z 1 - 4 ) , for w h i c h G M M -~ 1.08, a n d t h e c o r r e s p o n d i n g C E is z2(zl
2.25)(Zl - 4 ) ( z l - 1 / 2 . 2 5 ) ( z l - 1 / 4 ) ( z l
-
+ o, K z ~ ( z l
-
+ 0 . 1 7 4 ) ( z l + 5.734) 1 . 9 2 ) ( z l - 1 / 1 . 9 2 ) -- 0
O b v i o u s l y a f o u r t h o r d e r c o n t r o l l e r will be r e q u i r e d t o a c h i e v e t h i s c o m pensation. iii) F o r M --- 3, t h e e q u i v a l e n t p l a n t a f t e r z e r o r e l o c a t i o n w o u l d b e z l / ( z , 3 . 3 7 5 ) ( z l - 8 ) , for w h i c h G M M "- 1.16 a n d t h e c o r r e s p o n d i n g C E is
-
Z2(Zl -- 3 . 3 7 5 ) ( Z l -- 8)(z I 1 / 3 . 3 7 5 ) ( Z l -- 1 / 8 ) + o l K z ~ ( z l -- 2.46)(Zl -- 1 / 2 . 4 6 ) = 0 -
-
H e r e t h e c o n t r o l l e r o r d e r n e c e s s a r y is a p p a r e n t l y 2. T h i s value, h o w e v e r , d o e s n o t s a t i s f y e q n . ( 3 1 ) w h i c h r e q u i r e s t h e m i n i m u m c o n t r o l l e r o r d e r to be 3. So, a t h i r d o r d e r c o n t r o l l e r is to b e e m p l o y e d . B u t t h e t h i r d c o n t r o l l e r zero a n d p o l e m a y b o t h b e p l a c e d at o r i g i n so t h a t t h e a b o v e C E r e m a i n s essentially the same. iv) For M = 1, t h e b e s t G M of 1.22 is o b t a i n e d w h e n Ib[---. 1. F o r b ~ 1.5 or 2, h o w e v e r , t h e a t t a i n a b l e G M is o n l y 1. T h e n 3 - p e r i o d i c c o n t r o l l e r s will be t h e b e s t . Example
7 C o n s i d e r a g a i n a (4, 1 ) - p l a n t ( z - b ) / z ( z
+ x/-~)(z
2 - 2 z + 2).
i) M -- 1, 2 a n d 3 n e e d o n l y be c o n s i d e r e d . ii) F o r M -- 2, t h e p l a n t a f t e r zero r e l o c a t i o n w o u l d b e Z l / ( Z l for w h i c h G M M -- 1.56, t h e c o r r e s p o n d i n g C E b e i n g zl(zl
-
+ r
1.5)(Zl2 4 4 ) ,
1.5)(z~ + 4)(Zl - 1 / 1 . 5 ) ( z ~ + 1 / 4 ) ( z l + 1 . 5 ) ( z l + 1 / 1 . 5 ) - 2 . 1 4 ) ( z 1 - 1 / 2 . 1 4 ) ( z 1 + 2 . 1 3 ) ( z l + 1 / 2 . 1 3 ) -- 0
C l e a r l y a 5 t h o r d e r c o n t r o l l e r is r e q u i r e d to a c h i e v e t h i s c o m p e n s a t i o n . s u c h e v e n - i n p u t c o n t r o l l e r is
One
q o ( z ) = z ( 0 . 0 0 3 4 z 4 - 4 . 4 2 3 2 z 3 - 8 . 3 3 9 5 z 2 - 6 . 0 1 2 4 z - 1.3049) ql ( z ) "- q o ( z ) po(z)
-- z s + 1.8851z 4 + 2.4966z 3 + 2.506z 2 + 1.5455z + 0.4083
pl(~)
= 0
iii) F o r M - 3, t h e z e r o r e l o c a t e d p l a n t w o u l d be z ~ / ( z l + 1.837)(z~ + 4 z l + 8), for w h i c h G M M -- 1.313, a n d t h e c o r r e s p o n d i n g C E is
Z2(Zl + 1.837)(Zl2 + 4zl + 8 ) ( z I + 1 / 1 . 8 3 7 ) ( z ~ + 0 . 5 z l + 1 / 8 ) + c ~ K z ~ ( z 2 + 3 . 1 1 8 z l + 3 . 0 5 3 ) ( z ~ + 1.02zl + 0 . 3 2 7 ) - - 0
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
161
A f o u r t h o r d e r controller would be required to achieve this. iv) For M = 1, t h e best G M of 2.53 is o b t a i n e d when Ibl ---* 1. For b ---. - x / 1 . 5 , however, the GM will be 1, a n d t h e n t h e 2-periodic controller will be t h e best.
VII
Conclusions
and Discussions
T h e c o n t e n t s of this c h a p t e r can be s u m m a r i s e d as follows: 1. A linear, discrete, M - p e r i o d i c controller s t r u c t u r e t h a t has t h e m a x -
i m u m possible degrees of freedom for its o r d e r has been presented. 2. T w o a n a l y t i c a l m e t h o d s - the lifting technique and t h e Floquet theory approach ~ have b e e n p r e s e n t e d for dealing with periodic systems. It is seen t h a t while the f o r m e r is m o r e general in t h a t it can, besides analysing stability, yield t h e i n p u t - o u t p u t relation as well, the l a t t e r yields t h e C E in a m o r e s t r a i g h t f o r w a r d fashion. 3. Based on a s t u d y of t h e features of the closed loop C E of a L D T I p l a n t c o m p e n s a t e d by a M - p e r i o d i c controller, t h e zero p l a c e m e n t c a p a b i l i t y of periodic controllers has been investigated. It is seen t h a t while periodic controllers can relocate all the zeros of a n t h o r d e r bicausal plant, they can relocate only u p t o ( n - 1) zeros ( s o m e of which might originally have been at infinity) for n t h o r d e r strictly p r o p e r plants. 4. Finally the aspect of GM i m p r o v e m e n t of u n s t a b l e p l a n t s with N M P zeros via the zero p l a c e m e n t a p p r o a c h has been i n v e s t i g a t e d leading to t h e conclusions t h a t (a) 2-periodic controllers can provide infinite G M c o m p e n s a t i o n to bicausal plants, a n d (b) for o p t i m u m G M comp e n s a t i o n of a n-pole, r-zero strictly p r o p e r plant the p e r i o d i c i t y M of t h e controller should, in general, be a p r i m e n u m b e r not g r e a t e r t h a n m a x { 2 , (n - r)}. It m a y be n o t e d t h a t zero p l a c e m e n t is not the only way t h r o u g h which G M c o m p e n s a t i o n can be provided to a plant, in fact, in [2] p e r i o d i c controllers have been used to do t h e s a m e job via t h e factorization a p p r o a c h which, as such, is m o r e general t h a n t h e zero p l a c e m e n t a p p r o a c h . T h e p r o b l e m of o p t i m u m G M c o m p e n s a t i o n using the f a c t o r i z a t i o n a p p r o a c h , however, has not so far been solved in literature. (Some results r e g a r d i n g this are p r e s e n t e d in [11].) Besides GM i m p r o v e m e n t , t h e possibilities of using periodic controllers for d i s t u r b a n c e rejection and for s i m u l t a n e o u s stabilization have also been
162
SARIT K. DAS AND P. K. RAJAGOPALAN
investigated in literature. Regarding the former, a l t h o u g h it has been shown [2][3] t h a t periodic controllers offer no a d v a n t a g e over L D T I ones so far as u n i f o r m d i s t u r b a n c e rejection is concerned, the possibility t h a t such controllers may provide superior d i s t u r b a n c e rejection at certain r e g u l a r intervals of t i m e (say, the even instants) can not be ruled out. Further, regarding the latter, although it has been claimed in [2] t h a t generically a M - p e r i o d i c controller can not stabilize a set of M plants s i m u l t a n e o u s l y for M > 2, it has been shown in [12] t h a t M - p e r i o d i c controllers can not only stabilize but can as well place the poles of M plants s i m u l t a n e o u s l y for any M.
VIII
References
1. T. Kailath, L i n e a r S y s t e m s , Prentice Hall Inc., Englewood Cliffs, N.J. (1980). 2. P. P. K h a r g o n e k a r , K. Poolla, and A. T a n n e n b a u m , " R o b u s t control of linear time invariant plants using periodic c o m p e n s a t i o n " , I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l A C - 3 0 , pp.1088-1096 (1985). 3. J. S. S h a m m a , and M. A. Dahleh, " T i m e varying versus t i m e invariant c o m p e n s a t i o n for rejection of persistent b o u n d e d d i s t u r b a n c e s and robust stabilization", I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l A C 36, pp.838-847 (1991). 4. M. A. Dahleh, P. G. Voulgaris, and L. S. Valvani, " O p t i m a l and robust controllers for periodic and m u l t i r a t e systems", I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l AC-37', pp.1734-1745 (1992). 5. S. K. Das, and P. K. R a j a g o p a l a n , "Periodic d i s c r e t e - t i m e systems: stability analysis and robust control using zero p l a c e m e n t " , I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l A C - 3 7 , pp.374-378 (1992). 6. E. T. W h i t t a k e r , A c o u r s e in m o d e r n a n a l y s i s , C a m b r i d g e University Press, N.Y. (1962). 7. B. A. Francis, and T. T. Georgieu, "Stability theory for linear timeinvariant plants with periodic digital controllers", I E E E T r a n s a c t i o n s on A u t o m a t i c C o n t r o l A C - 3 3 , pp.820-832 (1988). 8. S. K. Das, and P. K. R a j a g o p a l a n , "Infinite d e t e r m i n a n t m e t h o d s for stability analysis of periodic s y s t e m s " , P r o c e e d i n g s I E E P a r t - D 131, pp.189-201 (1984).
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
163
9. A. T a n n e n b a u m , "Feedback s t a b i h z a t i o n of linear d y n a m i c a l plants with u n c e r t a i n t y in the gain factor", International Journal of Control 32, pp.1-16 (1980). 10. I. Horowitz, "Design of feedback s y s t e m s with n o n - m i n i m u m - p h a s e u n s t a b l e plants", International Journal of System Science 10, pp. 1025-1040 (1979). 11. S. K. Das, and P. K. R a j a g o p a l a n , " M - p e r i o d i c controller for maxim u m gain m a r g i n for N M P s y s t e m s " , P r e s e n t e d at S I A M Conference on Linear Algebra, San Francisco, Nov. 5-8, (1990). 12. A. K. Singh, "Periodic c o m p e n s a t i o n of discrete linear time invariant plants", Ph.D. Thesis, Indian I n s t i t u t e of Technology, K h a r a g p u r , India (1994). 13. C. A. Lin, and C. W. King, "Minimal periodic realizations of transfer m a t r i c e s " , I E E E Transactions on Automatic Control A C - 3 8 , pp.462466 (1993).
IX A
Appendices SISO
Periodic
Realization
of Transfer
Matrices
In Section III-A we have seen how the equivalent M x M transfer m a t r i x ,
G(z M), c o r r e s p o n d i n g to a M - p e r i o d i c . s y s t e m can be o b t a i n e d . Here we a t t e m p t the reverse p r o b l e m of obtaining the minimal, SISO, M - p e r i o d i c realization of a given M x M C~ satisfying the causality condition (~(oo) lower triangular. It will, however, be seen t h a t such a m i n i m a l realization does not utilise all its possible degrees of freedom. Noting t h a t for a 2 x 2 transfer m a t r i x C~(z2) the (1, 1)-element, G l l , s t a n d s for the even instant input to the even instant o u t p u t t r a n s f o r m a t i o n , and, similarly, G12, G21 and G22 stand, respectively, for the odd to even, even to odd and odd to odd t r a n s f o r m a t i o n s , G may as such be realized by realizing each GU, i , j = 1, 2, individually with the a p p r o p r i a t e input signal applied to it and its o u t p u t taken at the correct instants. Fig.4 shows this realization. (Note t h a t for causality the o u t p u t s of the blocks G12 and G21 must be delayed by one sampling period.) It is clear t h a t for a M x M G(z M) to be realized in this fashion one would require (in the worst case) M2rn M - d e l a y elements (i.e., 1/z M blocks) where rn is the order of t h e c o m m o n d e n o m i n a t o r d(z M) of the elements of (~. However, it a p p e a r s logical t h a t the s a m e should be realizable using only as m a n y delay e l e m e n t s as is the highest degree of z occurring in d (i.e., M r n ) provided, of
164
SARIT K. DAS AND R K. RAJAGOPALAN
y,(.,~)
U~(~ 2) "-~ (z~)
,
!
U
J_l] 7 z
J .
I
--1
7 ~
z - 1 r.,ro(z 2 )
7 ,G"(z
)
_/
"x.
z - ~ Y o ( z 2)
F i g u r e 4: D i r e c t r e a l i z a t i o n of a 2 • 2 t r a n s f e r m a t r i x
course, t h e g a i n s are now allowed to b e c o m e p e r i o d i c . W e s h o w n e x t t h a t such i n d e e d is t h e case. F i r s t we b r i n g o u t an i m p o r t a n t p r o p e r t y of t h e G c o r r e s p o n d i n g to a c a u s a l p e r i o d i c m a p ( b e s i d e s t h e one t h a t G ( o o ) is lower t r i a n g u l a r ) [12]. 3 Given a causal, SISO, M-periodic system, if one constructs the corresponding M x M L D T I transfer matriz G ( z M) without performing any (possible) pole-zero cancellation at origin in its elements and ezpresses the 9~m~ ~ ~ ( ~ ) = ~(zU)/d(~), ~ h ~ d i~ th~ t.~.m, olth~ a r of G and 1V is a polynomial matriz, then, apart from G(oo) bein 9 lower triangular, fit(O) would be upper triangular. Conversely, if.for a given G with
Lemma
G(o~) tow~,, t,.ia,~g,,u,,, th~ .,~(o) i, . o f ,,pp~,. t,.i,,ng,,h,,., th~n it sig,,,i/ie~ that the system has a hidden mode at origin not reflected in d. Proof: For c o n v e n i e n c e we p r o v e t h e r e s u l t for M = 2. T h e p r o o f can easily be e x t e n d e d for M > 2. It was s h o w n in S e c t i o n I I I - A t h a t t h e even a n d o d d o u t p u t s of a c a u s a l 2 - p e r i o d i c m a p are r e l a t e d to t h e even a n d o d d i n p u t s by t h e t r a n s f e r m a t r i x G as given by e q n . ( 7 ) or, a l t e r n a t i v e l y , by t h e m a t r i x r as given by e q n . ( 8 ) . N o t e , however, t h a t of t h e two m a t r i c e s r a n d r G is m o r e f u n d a m e n t a l b e c a u s e it r e l a t e s t h e even a n d o d d i n p u t a n d o u t p u t s i g n a l s in t h e c o r r e c t fashion. In o t h e r words, given a 2 - p e r i o d i c m a p , its 2 • 2 L D T I r e p r e s e n t a t i o n can p h y s i c a l l y be o b t a i n e d o n l y in t h e f o r m G, a n d , conversely, a given 2 • 2 m a t r i x of t h e form of r m u s t first be t r a n s f o r m e d to t h e form r if it is to b e p h y s i c a l l y realized as a 2 - p e r i o d i c s y s t e m . Let d'(z 2) be t h e l.c.m, of t h e d e n o m i n a t o r s of Gij(z2), i , j = 1,2. Now, d e p e n d i n g on w h e t h e r G i j , i ~: j , h a v e n u m e r a t o r f a c t o r s of z 2 or n o t , four cases m a y arise:
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
165
1. Neither G12 nor G21 has a n u m e r a t o r .factor of z 2" T h e n f r o m eqns. ~,
(7) a n d (8) t h e c o m m o n d e n o m i n a t o r forms for G a n d G b e c o m e
[ z2Nll G -z N21
(~ --
z Nil z 2 N2i
zN12 ] z 2 N~2
1
N12 ] 1 z 2 N22 Z 2d'
1V d
w h e r e Nij --- G q d ' a n d d e g ( N i j ) <_ d e g ( d ' ) .
2. G21 has a n u m e r a t o r factor of z 2 but G12 does not: T h e n (~=
, z 3 N2i
(~._ [ z 2 N i i z 4N~i I
2z d ,
z2N2:
Ni2 ] 1 z 2N22 z 2d' --
/~r d
2
w h e r e N21 := N 2 i / z . 3. G12 has a n u m e r a t o r factor of z 2 but G21 does not: T h e n G=
z ~ N12 ' 1
[Z2Nll
zN~l G -
z2N22
Ni2
z2Nll z2N:l
z2N22
1
z2d ' z2d '
d
w h e r e N~2 "= N t 2 / z ~. N o t e t h a t we have not cancelled t h e f a c t o r z 2 from t h e n u m e r a t o r s and t h e d e n o m i n a t o r s of G b e c a u s e such a c a n c e l l a t i o n does not t a k e place in t h e f u n d a m e n t a l f o r m G.
4. Both Gt2 and G2t have the numerator :factor of z 2" T h e n Nil z g~t Nit 2 N~ t
zN~2 ] 1 N22 "-~ N~2 ] 1 N22 d'
IV d
N o t e t h a t unlike t h e p r e v i o u s t h r e e cases here d ---- d'. O b s e r v e now t h a t in all t h e four cases possible, (~(c~) is lower t r i a n g u l a r a n d /~r(0) is u p p e r t r i a n g u l a r . T h i s establishes t h e first p a r t of t h e L e m m a . T h e converse p a r t also b e c o m e s e s t a b l i s h e d if one n o t e s t h a t if, for a given /~r, /~r(0 ) is n o t u p p e r t r i a n g u l a r t h e n t h a t m u s t c o r r e s p o n d to t h e case 3 n o t e d a b o v e w i t h t h e pole-zero c a n c e l l a t i o n at origin effected. I-1 T h e c o m m o n d e n o m i n a t o r r e p r e s e n t a t i o n of a 2 x 2 G m a y now be used to o b t a i n a 2m o r d e r r e a l i z a t i o n as follows:
166 Theorem
SARIT K. DAS AND E K. RAJAGOPALAN
7 Given a 2 x 2 transfer matrix G(z 2)-
[l~U(z2)]ld(z2),
i,j--
1,2
"4-'''
4- a l z
(41)
wh ere
2.~
"~- a m - 1 z
2(m-t)
d ( z 2)
=
z
)~ri/(z2)
--
bij,rnz 2m -}- b i j , m - i z 2 ( m - l ) -b . . . 9..-t- bij,lZ 2 ~- bij,o,
(42)
~- a 0
i, j -- 1, 2
(43)
with bx2,rn = b2x,o = 0 ( s o as 1o satisfy the causality conditions of L e m m a 3), a 2-periodic 2 m - o r d e r realization (having input u and output y ) of the s a ~ r L e iS
y , ( N + 1)
=
yi+I(N),
i= 0,1,...,(2m-1)
(44)
m--1
y:~(N)
=
u(N)-
~
aiy2i(N)
(45)
+ (-1) Ndi,1]yi(N)
(46)
i--O 2rn
y(N)
=
~[d,,o i--O
wh ere
d2i,1 i " = d:, +~,od~,==,o(b;1,~ (b~,, (b~:,,+- b22,i)/2 b~2, + b:~,,+~)/2 ,)/2
i=
d2i+1,1 -- (b12,i -- b21,i+1)/2
[
O, 1 , . . . , m 0,1 . . . ( m - ' '
(47)
1) J
( F i g . 5 s h o w s t h i s r e a l i z a t i o n for m = 2.)
Proof: It follows f r o m e q n s . ( 4 4 ) a n d (45) t h a t yo(N) = d-;(z2)u(N) F u r t h e r , e q n . ( 4 6 ) y i e l d s t h e r e l a t i o n b e t w e e n y a n d yo as
y ( N ) = [qo(z) + ( - 1 ) ~ q l ( z ) ] y o ( N ) where
qo = d2,~,oz 2m + d2,~-l,oz ql = d2,~,;
Z2m
+ d2,,~_;,1
2m -- 1
Z2m--1
+'"
+ do,o
+--.
+ do,;
It w a s s h o w n in t h e p r o o f of T h e o r e m 1 t h a t t h e 2 x 2 L D T I m a t r i x , (~1, c o r r e s p o n d i n g to t h e t i m e v a r y i n g m a p b e t w e e n Y0 a n d y is as g i v e n by e q n . ( 1 2 ) . H e n c e we h a v e t h e o v e r a l l t r a n s f e r f u n c t i o n as ( ~ ; / d ( z 2) w h i c h is
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL
d4o 4-
(--i) Nd41
167
!
d30 + (-- 1)N d317
d,0 +
I
, ,
F i g u r e 5: M i n i m a l 2-periodic r e a l i z a t i o n of a 2nd o r d e r 2 x 2 t r a n s f e r m a t r i x
of t h e s a m e form as eqn.(41). Now, it is r o u t i n e to show t h a t for G1 to be e qu al to N~j as given by eqn.(43), t h e periodic feedforward gains d~j,o and dii,1 m u s t be as given by eqn.(47). [-I It is not difficult to generalize T h e o r e m 7 an d see how a M x M t r a n s f e r m a t r i x can be realized using a M m - o r d e r , M - p e r i o d i c r e a l i z a t i o n . N ote in p a r t i c u l a r t h a t t h e n u m e r a t o r m a t r i x in such a case would involve ( M 2 r n + M ) a r b i t r a r y c o n s t a n t s , a n d t h a t t h e n u m b e r of a s s i g n a b l e coefficients available in t h e form of t h e M - p e r i o d i c f eed f o r w ar d gains is also the same. In fact, for a given n u m e r a t o r m a t r i x t h e gains will have u n i q u e values. It follows from t h e r e s u l ts o b t a i n e d in [13] t h a t for a g e n e r a l t r a n s f e r m a t r i x G of t h e form of eqn.(41), t h e m i n i m u m o r d e r r e a l i z a t i o n would involve 2m s t at es . Thus the realization obtained above is, f o r an arbitrary G, a m i n i m a l one. (In some p a r t i c u l a r cases, however, (~ m a y have lower, even m - o r d e r , r e a l i z a t i o n s [13].) Notice in this c o n t e x t t h a t t h e above r e a l i z a t i o n m a k e s use of only rn of t he M m feedback loops available, t h e rest being identically e q u a l to zero. (Even t h e rn feedback loops t h a t are used have only t i m e - i n a v a r i a n t gains.) Hence it is o b v i o u s l y n o t a m a x i m u m - d e g r e e s - o f - f r e e d o m c o n f i g u r a t i o n . B
Completeness
of the
Controller
Form
Assumed
To show t h a t t h e c o n t r o l l e r shown in Fig.2 has t h e m a x i m u m p o ssi b l e degrees of freedom, it suffices to show t h a t any a d d i t i o n a l p e r i o d i c gain
168
SARIT K. DAS AND E K. RAJAGOPALAN
i n t r o d u c e d in t h e c o n t r o l l e r can be a b s o r b e d in t h e existing gain p a r a m e t e r s C~(N) and D ~ ( N ) t h e m s e l v e s . Now, to e s t a b l i s h this it is sufficient to show t h a t a p e r i o d i c gain i n t r o d u c e d at a point p r e c e d i n g a z -1 delay block can be t r a n s f e r r e d to a point following t h e same. To this end we h av e t h e following result. Lemma
4
z - - l ( f 0 + fiN'/1 -I" ~ 2 N f 2 4" "'" "+" o~(M--I)N f M - 1 ) = (fO "}- o t N - l f l "}" f f 2 ( N - 1 ) f 2 + o~(M--I)(N--1)fM--1)Z--I
where c~ is as given by eqn.(4) and fi, i = 0 , . . . , ( M Fourier coej~icients of a M-periodic function f ( N ) .
1), are the discrete
Proof: Itis easy to verify t h a t t h e expression in t h e left h a n d side p r o d u c e s , while o p e r a t i n g on a signal z ( N ) , t h e s a m e o u t p u t as t h e e x p r e s s i o n in t h e right. !"! Using this r es u lt one m a y c o n t i n u e to shift any p e r i o d i c gain i n t r o d u c e d in t h e f o rward p a t h till it is c o m p l e t e l y a b s o r b e d in t h e Ci's a n d Di's.
C
GM
C o m p e n s a t i o n using L D T I C o n t r o l l e r
In this section we p r e s e n t briefly two m e t h o d s for finding t h e m a x i m u m G M c o m p e n s a t i o n possible for a given L D T I p l a n t using L D T I controllers. 1
The Nevanlinna-Pick
(NP)
Matrix
Method
[9]
T h i s m e t h o d is b a s e d on t h e N e v a n l i n n a - P i c k i n t e r p o l a t i o n results. T h e c o m p u t a t i o n a l s t e p s n e c e s s a r y for finding t h e m a x i m u m G M o b t a i n a b l e following this a p p r o a c h are: i) O b t a i n t h e s - p l a n e e q u i v a l e n t of t h e given p l a n t using t h e b i l i n e a r t r a n s f o r m a t i o n z = (8 + 1 ) / ( 8 1). Let t h e e q u i v a l e n t p l a n t be b ( 8 ) / a ( 8 ) . ii) F a c t o r t h e s - p l a n e p l a n t as
b(,)/~(,) =
[b~(,)lax(,)]lb(,)la(,)]
where t h e r o o t s of bl a n d al lie in t h e left h a lf and t h o s e of b an d a lie in t h e right half of t h e c o m p l e x 8-plane. (Single r o o t s on t h e i m a g i n a r y axis m a y be c o n s i d e r e d to be in t h e left half plane, b u t such m u l t i p l e r o o t s are to be c o n s i d e r e d as right h a l f ones.) Also let
7' b(~
= ]-[(~
i----1
-
~,)
TECHNIQUES OF ANALYSIS AND ROBUST CONTROL iii) Defining q ( s ) : = a ( s ) / a ( - s ) ,
169
obtain
q , ( , ) - - q(s)l.=#, ,
i=
1,2,...,r
iv) C o n s t r u c t the N P m a t r i x given by
o,=
1 - 1,712/q, qj
3, + aj
] ,a=t.2,...,,
where the u p p e r bar denotes complex c o n j u g a t e operation. v) Find the largest n u m b e r Yop < 1 such t h a t for 7/ = yop or 1/yop the m a t r i x Q is positive definite. (Note t h a t Q = Q* where the * d e n o t e s complex c o n j u g a t e t r a n s p o s e operation. Hence the positive definiteness of Q can be verified by checking only its principal minors.) T h e n the m a x i m u m GM o b t a i n a b l e is given by GMrr,,= = (1 + ~/op)2)/(1 -- flop) 2
Note t h a t this a p p r o a c h does not yield t h e controller directly. 2
The Horowitz Approach
[10]
T h i s approach, which is based on the s y m m e t r y of t h e root loci of the c o m p o s i t e s y s t e m a b o u t the jw-axis, yields not only GMrr, ox but the c o r r e s p o n d i n g controller as well. T h e steps involved (for 8-domain plants) are as follows: i) Consider only the right half plane poles (n in n u m b e r ) and the right half plane zeros (r in n u m b e r ) of the given plant. Let this collection be given by G(-8) = b(-8)/a(-s). Now, the controller H should b e s u c h t h a t the loop transfer function is s y m m e t r i c a l a b o u t the jw-axis, i.e., G H = K ~ ( s ) ~ ( - s ) , with &(s) having (n + r - 1) poles as well as zeros. Let ~b(s) have only left half plane poles and zeros and choose it as
~(,)
B(s) =
b ( s ) ( s " - t + B , _ 2 s " - 2 + . . " + Bo) =
Z(,)
~(o)(o'-~ + A,_~o "-~ + . . . + ao)
ii) S e p a r a t e B ( , ) and A(a) into monic even and odd p a r t s as follows: B ( s ) = c t e t ( s ) + dtol(8)
A ( , ) = ~2e~(0) +
d~o~(,)
Obviously, if A(8) and B(8) are monic, we would have either ct -- c2 -- 1 o r d t = d 2 = 1. iii) Make or(a) = o~(8) and et (8) = e2(s) by e q u a t i n g coefficients. T h i s will
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SARIT K. DAS AND P. K. RAJAGOPALAN
give rise to (n + r obtain A0,...,Ar_2
2) e q u a t i o n s in p r e c i s e l y as m a n y u n k n o w n s . S o l v i n g and B0,...,B,-2. T h e n t h e c o n t r o l l e r will be given
by H(8)= ~b(8)~b(-s)/G(-8).
iv) G M r n a x = ( d r ~ d 2 ) 2 or ( c l / c 2 ) ~ w h i c h e v e r is different f r o m 1. ( N o t e t h a t if G M c o m e s o u t to be less t h a n unity, t h e n inverse of t h a t n u m b e r is t o be considered.)
R O B U S T FAULT D E T E C T I O N A N D ISOLATION ( F D I ) S Y S T E M S R. J. Patton University of Hull Department of Electronic Engineering Cottingham Road, HULL HU6 7RX, UK J. Chen University of Strathclyde Division of Dynamics and Control Glasgow, G1 1XJ, UK
I. I N T R O D U C T I O N The model-based approach to fault detection and isolation (FDI) in automated processes has received considerable attention during the last two decades, both in a research context and also in the domain of application studies on real processes [1-7]. There are a great variety of methods in the literature, based on the use of mathematical models of the process being monitored for faults. The procedures of using model information to generate additional signals, to be compared with the original measured quantities, is known as analytical redundancy, which can be Used to obviate the repetition of hardware in the alternative approach known as hardware redundancy. Analytical redundancy is a form of dissimilar redundancy, in the following sense. Given one measurement from the dynamical system under control, a mathematical model of the system is used to generate estimates of other measurements, thereby developing redundancy in an analytical form [8]. Models used for FDI are usually linear, in either the time or frequency domains, however non-linear methods have also been used based upon the known non-linear differential equations of a physical system [6,9]. The most common approach to model-based FDI is the use of state estimation observers or Kalman filters [5,9], which make direct use of a CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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state-space model of the system being monitored. There are, however a number of other important methods, which whilst based upon the use of a mathematical model of the system, do not use the feedback structure of a state observer [7,10,11]. In the framework of model-based fault diagnosis, faults are detected by setting a (fixed or variable) threshold on each residual signal. A number of residuals can be designed, each having special sensitivity to individual faults occurring in different locations in the system. The subsequent analysis of each residual, once a threshold is exceeded, then leads to fault isolation [5]. Monitored System
Reference Input Controller
Faults ~ Input Actuators
: ~
Residual
~.'1,,
Generation
"
~ Output
Plant Dynamics
" Sensors
Residuals J Decision q Making
Fault Diagnosis System
1
Fault Information
Figure 1" General structure of model-based fault diagnosis Fig. 1 illustrates the general concept of the structure of model-based fault diagnosis system comprising two main stages of residual generation and decision making [7,10]. Model-based FDI approaches are built upon a number of idealized assumptions, one of which is that the mathematical model used is a faithful replica of the plant dynamics. This is, of course, not possible in practice, as an accurate and complete mathematical description of a process is never available. Sometimes the mathematical structure of the dynamical system is not fully known. For other applications, the parameters of the system may not be fully known or may only be known over a limited range of the plant's operation. There is therefore always a "model-reality mismatch" between the plant dynamics and the model used for FDI. Another idealizing assumption accompanying the model-based approach to FDI is that the
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true characteristic (e.g. spectral shape, variance, mean, stationarity, etc) of disturbance and noise signals acting upon the system are known. This can also never be true in practice. As the complexity of a dynamical system increases, the harder the task of modelling the system and its disturbances and we can speak of an uncertain system, for which there is an uncertainty of knowledge of the system's structure, parameters and effect of disturbances. There are therefore robustness problems in FDI with respect to uncertainty and disturbances [3,5-7,11-16]. Whilst, these can be considered as separate problems, it is customary to lump all uncertain effects together as far as is possible and we thus speak of a general robustness problem in FDI, in which the goal is to discriminate between the effects of faults and the effects of uncertain signals and perturbations in the system. The uncertainties can of course be quite normal effects, arising for example, as a consequence of a non-linear system. The robustness problem in FDI is thus defined as the maximization of the detectability and isolability of faults together with the minimisation of the effect of uncertainty and disturbance on the FDI procedure [11-16]. The optimization problems can be achieved using sensitivity theory, as long as due care has been paid to the robustness of the global system operation. Robustness in FDI has been recognised as a professional field for a considerable time [11-20]. Some of the earlier approaches were really based on local sensitivity requirements, rather than producing truly robust solutions. However, during the past 15 years, there has been increasing activity focused on the development of truly robust methods for FDI. Most of these studies focus on the problem of creating robustness in residual generation [5,14,17,19]. Other proposals, focus on the problem of making the actual decision-making stage of FDI [13,14,20]. The ultimate goal of robustness is to provide rapid and reliable detection and isolation of system faults when the plant under control is disturbed, and when the mathematical model upon which the diagnosis is based cannot faithfully reproduce the full dynamic operation of the plant. The aim of this chapter is to review methods for robust fault diagnosis, based principally on residual generation. Some of the key challenges and potential for future directions in the research are drawn up. In order to summarize the different contributions to achieving robustness in FDI, Sections II and III describe the essential properties of residual generation and fault isolation. In Section IV the need for robustness in FDI is outlined, from both time domain and frequency domain perspectives. Section V outlines the robustness problems with respect to disturbances (or structured uncertainty). Section VI reviews the approaches for robust FDI for systems with modelling errors (unstructured uncertainty). Section VII presents some dis-
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cussions on frequency domain robust FDI and Section VIII studies robust decision-making based on adaptive threshold or fuzzy logic. Finally, Section IX provides some pointers as to the likely developments in the field and stresses the topics which are probably worthy of the most research activity in the future. II. MODEL-BASED A. S Y S T E M
METHODS
FOR
FDI
DYNAMICS
The state space model of for the system with faults is given by [21]:
( x(t) y(t)
-
Ax(t) + Bu(t) + R l f ( t ) Cx(t) + Du(t) + R2f(t)
(1)
where x(t) E IR n is state vector, f(t) E lit g is a fault vector, each element fi(t) (i = 1 , 2 , - . - , g ) corresponds to a specific fault. From a practical point of view, it is unreasonable to make further assumptions about the fault characteristics but consider these as unknown time functions. The matrices R1 and R2 are known as fault entry matrices which represent the effect of faults on the system. The vector u(t) E IR r is the system input and the vector y(t) E IR m is the output, and both vectors are known for FDI purpose. A, B, C and D are system model matrices with appropriate dimensions. This system model is also illustrated in Fig. 2.
R1
u(t)
Rz
x(t)
y(t)
Figure 2: General structure of model-based fault diagnosis An input-output transfer matrix representation for a system with possible
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faults is then described as"
(2)
y(s) -- Gu(s)u(s) + Gy(s)f(s) where
Gu(s)
-
Gy(s)
-
C(sI-A) -1B+D C ( s I - A ) - I R , + R2
(3)
The general model for a faulty system described by Eq.(1) in the timedomain and by Eq.(2) in the frequency-domain has been widely accepted in the fault diagnosis literature. B. G E N E R A L
STRUCTURE
OF R E S I D U A L
GENERATORS
The core element of model-based FDI is the generation of residuals. To provide useful information for FDI, each component of the residual vector r(t)(e IRp) generated should be insensitive to (independent f r o m ) s y s t e m inputs and therefore should satisfy the condition: r(t) -~ 0
if f
f(t) r 0
(4)
Eq.(4) is the basis for fault detection using model-based residuals. We now give a general definition of the residual signal (vector) in FDI as a linear function of inputs and outputs of the monitored system which is also independent of the normal operating state of the system. In the fault-free case, the residual is near to zero (very small in some sense). When a fault occurs in the system, the residual will grow significantly. This processor can be worked out in continuous time or can be a discrete-time algorithm in non recursive or recursive form. In the case of a non-linear process, model linearisation around an operating point should be considered, although nonlinear models can also be used for residual generation [22-24], according to the non-linearities of the plant. A general structure for all residual generators is shown in Fig.3. This structure is expressed mathematically as:
[ V(*) ] -
r ( s ) - [H~(s)
(5)
Here, Ha(s) and Hy(s) are transfer matrices which are realizable using stable linear systems. According to the definition, the residual is designed to become zero for the fault-free case and nonzero for faulty cases, i.e." r(t) - 0
iff
f(t)
0
(6)
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Figure 3: The general structure of a residual generator To satisfy this condition, the transfer matrices the constraint condition: +
H~,(s) and Hy(s)
must satisfy
(7)
= 0
Eq.(5) is a generalized representation of all residual generators [7]. The design of the residual generator results simply in the choice of the transfer matrices H,(s) and Hy(s) which must satisfy Eq.(7). The various ways of generating residuals correspond to different parameterisation of H,,(s) and Hy(s). One can obtain different residual generators using different forms for H~(s) and Hy(s). Using the design freedom, the desired performance of the residual can be achieved by suitable selection of H~(s) and Hy(s) [7,21]. A fault can be detected by comparing the residual evaluation function J(r(t)) with a threshold function T(t) according to the test given in below:
J((r(t)) J((r(t))
<_ T(t) > T(t)
for for
f(t) f(t)
- 0 ~s 0
If this test is positive (i.e. the threshold is exceeded by the residual ation function), we can hypothesize that a fault is likely. There are ways of defining evaluation functions and determining thresholds. example, the residual evaluation ' function can be chosen as a norm
evalumany As an of the
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residual vector and the threshold can be chosen as a constant positive value (fixed threshold). C. F A U L T D E T E C T A B I L I T Y When faults occur in the monitored process, the response of the residual vector is: g
r(s) - Hy(s)G/(s)f(s) - G~:(s)f(s) - ~[G~l(s)]~f~(s)
(8)
i=1
where Grf(s) = Hy(s)Gl(s) is defined as a fault transfer matrix which represent the relation between the residual and faults, [Grf(s)]i is the ith column of the transfer matrix G~l(s) and fi(s) is the ith component of f(s). The fault transfer matrix plays an important role in FDI and must be examined in detail [21,25]. In order to detect the ith fault f~ in the residual r(s), the ith column [Grl(s)]i of the transfer matrix Gr/(S) should be nonzero:
# 0
(9)
If this condition holds true, the ith fault fi is detectable in the residual r. This is defined as the fault detectability condition of the residual r to the fault fi. It is believed that the fault detectability alone is not enough to achieve reliable fault detection [21,25]. Hence, the strong fault detectability is introduced here as: [G~](O)]i 7~ 0 (10) If this condition is satisfied, we define that the ith fault fi is strongly detectable in the residual r. This condition can also be defined as the strong fault detectability condition of the residual r to the fault fi. If the frequency band of a certain fault is available, the residual can be designed maximally sensitive to this fault by frequency-shaping. This can be done by maximizing the following criterion: inf s We[W1,032]
(jw)}
(11)
where ~r{.} denotes the minimal singular values, and [wl,w2] denotes the frequency range in which the fault is most likely to occur. D. F A U L T I S O L A T I O N The successful detection of a fault is followed by the fault isolation procedure which will distinguish (isolate) a particular fault from others. Whilst a
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single residual signal is sufficient to detect faults, a set of residuals (or a vector of residuals) is usually required for fault isolation [5,9]. If a fault is distinguishable from other faults using one residual set (or a residual vector), it can be said that this fault is isolable using this residual set (or this residual vector). If the residual set (or the residual vector) can isolate all faults, we can then say that the residual set (or the residual vector) has the required isolability property. S t r u c t u r e d r e s i d u a l s set One approach to fulfil the fault isolation task is to design a set of structured residuals. Each residual is designed to be sensitive to a subset of faults, whilst remaining insensitive to the remaining faults. The residual set which has the required sensitivity to specific faults and insensitivity to other faults is known as the structured residuals set [11,21]. The design procedure consists of two steps, the first step is to specify the sensitivity and insensitivity relationships between residuals and faults according to the assigned isolation task, and the second is to design a set of residual generators according to the desired sensitivity and insensitivity relationships. The advantage of the structured residual set is that the diagnostic analysis is simplified to determining which of the residuals are non-zero. The threshold test may be performed separately for each residual, yielding a Boolean decision table, and the isolation task can be fulfilled using this table. If all possible faults are to be isolated, a residual set can be designed according to the following fault sensitivity conditions:
ri(t) = R(fi(t));
i E {1,2,...,g}
(12)
where R(.) denotes a functional relation. This is called as a dedicated residual set which is inspired by the dedicated observer scheme proposed by Clark [26]. A simple threshold logic can be used to make decision about the appearance of a specific fault by the logic decision according to:
ri(t) > 7~
==v
fi(t) r O;
iE{1,2,...,g}
where T/ (i = 1 , . . . , g ) are thresholds. This isolable residual structure is very simple and all faults can be detected simultaneously, however it is difficult to design in practice. Even when this structured residual set can be designed, there is normally no design freedom left to achieve other desirable performances such as robustness against modelling errors [5,27]. A most commonly used and better scheme in designing the residual set is
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to make each residual sensitive to all but one fault, i.e.
rl(t)
-
R(f2(t), . . ., fg(t))
ri(t)
-
R(fl(t),...,fi_l(t),fi+l(t),
,.,(t)
-
,fg(t))
(13)
.
This is defined as a generalized residual set. If all residuals of the generalized residual set are generated using a bank of observers (observer-based residual generators), the structure is known as the generalized observer scheme [5]. The isolation can again be performed using simple threshold testing according to the following logic:
r (t) <
rj(t) > Tj
Vj ~: i
}~
fi(t) ~: O; for i - 1 , 2 , . . .
,g
As a simple example, we will isolate three different faults {f l, f2, f3} by designing a residuals set {rl, r2, r3} using the following two methods" Method 1 f~ f2 f3
rl
r2
r3
1 0 0
0 1 0
0 0 1
f~ f2 f3
Method 2 rl r2 0 1 1 0 1 1
r3 1 1 0
S t r u c t u r e d R e s i d u a l Sets
In the tables above, a "1" in ith row and jth column denotes that the residual rj is sensitive to the fault fi, whilst a "0" denotes insensitivity. Faults can be uniquely isolated using either of the above methods. Fixed direction residual vector An alternative way of enhancing the isolability of faults is to design a directional residual vector which lies in a fixed and fault-specified direction (or subspace) in the residual space [11,28-30], in response to a particular fault. This is to make"
r(t l f i ( t ) ) -
ai(t)li;
i E {1,2,...,g}
(14)
where the constant vector li is the signature direction of the ith fault in the residual space and c~i is a scalar that depends on the fault size and dynamics. With the fixed directional residual, the fault isolation problem
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is one of determining which of the known fault signature directions the generated residual vector lies the closest to. To isolate faults reliably, each fault signature has to be uniquely related to one fault. Fig. 4 illustrates this fault isolation approach using a directional residual vector in which the residual is closed to 11 and the fault is most likely associated with the direction 11.
Fault direction 3 13~
'//
Fault direction 2
Fault direction 1 Figure 4: Directional residual vector for fault isolation
III. RESIDUAL
GENERATION
METHODS
Based on the definition of the linear generalized residual generator, a number of residual generation methods can be derived, using, for example: (a) Observer-based methods [5,6,7,9,26] 9 Full-order state observers 9 Generalized observers 9 Unknown input observers 9 Failure detection filters [28-30] 9 Kalman Filter and modifications [1,31,32] (b) Parity relation methods [10,11,19,32] (c) Stable factorisation of transfer matrices [20,34,35] (d) Parameter estimation [2,36] The above constitute the main approaches to model-based FDI using residual generation. It must be pointed out that most residual generation approaches are applicable for both continuous and discrete system models,
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however some approaches can only work for discrete models. In this chapter, if the continuous model is used, it implies that the technique can be applied to both continuous and discrete model, otherwise the technique is only applicable for discrete model. A. O B S E R V E R - B A S E D
APPROACHES
The basic idea behind the observer or filter-based approaches is to estimate the outputs of the system from the measurements (or a subset of measurements) by using either Luenberger observer(s) in a deterministic setting or Kalman filter(s) in a stochastic setting. Then, the (weighted) output estimation error (or innovations in the stochastic case), is used as a residual. The flexibility in selecting observer gains has been fully exploited in the literature yielding a rich variety of FDI schemes, the most recently development can be found in various survey papers: e.g. [5-9,13,14,22]. What we are interested in FDI is the estimation of outputs using an observer, whilst the estimation of the state vector is unnecessary. Indeed, a functional observer is suitable for this task. In practice, the order of the functional observer is less than the order of a state observer. It is desired to estimate a linear function of the state, i.e. Lx(t), using a functional (or generalized) Luenberger observer with the following structure: { ~(t)
-
F z ( t ) + Ky(t) + Ju(t) z(t) + Ry(t) + s (t)
(15)
where z(t) E IR q is the state vector of this functional observer, F, K, J, R, G and S are matrices with appropriate dimensions. The output w(t) of this observer is said to be an estimate of Lx(t), for the system described in Eq.(1), in an asymptotic sense if in the absence of faults: lim [ w ( t ) - Lx(t)] = 0
(16)
t---, cx:~
To introduce a transformation matrix T, the observer shown in Eq.(15) will generate the estimate Lx(t) in the asymptotic sense if and only if the following conditions hold [37]: F has stable eigenvalues TA - FT = K C J=TB-KD RC + GT = L S+RD=O
(17)
The necessary and sufficient condition for the existence of the observer given by Eq.(15) for the system Eq.(1) is that the pair ( C , A ) is observable [37].
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R.J. PATTONAND J. CHEN
In order to generate residuals, we need to estimate the system output. If we assign: L=C (18) We have the output estimation as:
~)(t) = w(t) + Du(t)
(19)
The residual vector r(t) is defined as:
r(t) = Q [ y ( t ) - y(t)] = L1 z(t) + L2y(t) + L3u(t)
(20)
where: L1
L2 L3
-
-QG QR - Q ( S + D)
Q -
Now, the residual generator based on a generalized Luenberger is illustrated in Fig. 5 and given by the following equation: { i(t)
-
r(t)
-
F z ( t ) + K y ( t ) + Ju(t) Llz(t) + L2y(t) + L3u(t)
(21)
And the matrices in this equation should satisfy the following conditions: F has stable eigenvalues
TA- FT- K C J-TB-KD L1T -I- L2C - 0 L3 + L2 D - 0
(22)
The Laplace transformation of the residual is thus:
r(s) = [ L l ( s I - F ) - I K + L2]y(s) + [ L l ( s I - F ) - I J + La]u(s)
(23)
The residual generator based on a generalized Luenberger observer is shown in Fig. 5. It can be seen that there is a feedback structure imbedded within it. The feedback can be used to improve the dynamic behaviour of residuals. When we apply the residual generator described by Eq.(21) to the system described by Eq.(1), the residual will be: { ~(t)
-
r(t)
-
Fe(t)- TRlf(t) + KR2f(t) Lie(t)+L2R2f(t)
(24)
where e(t) = z ( t ) - Tx(t). It can be seen that the residual depends solely and totally on faults.
ROBUSTFDI SYSTEMS
183
fault I f(t) input
output
J
-1 System
A
y(t)
f
I 11 z(t)
.I
7L1
+ +.
residual w
r(t)
A
'7L3|
k.
Figure 5: Residual generation via a generalized observer The simplest method in observer-based residual generation is to use a full order observer, in this case the observer dimension q equals n and we h ave:
F
-
A-KC
L2
-
-Q
J
-
B-
L3
-
QD
KD
Hence, the transfer function matrices for a full-order observer based residual generator are given by: Hu(s H~(s)
-
Q { C [ s I - ( A - K C ) ] - I K - I} Q { C [ s I - (A - K C ) ] - I ( B - It'D) + D}
(25)
To alter the frequency response of the residual, the residual weighting matrix Q can be changed into a dynamic weighting Q(s). For any dynamic system, the observer-based residual generator always exists. This is because any input-output transfer function matrix has the observable realization. That is to say, the output estimator always exists although a suitable state observer cannot always be designed. To isolate faults, the observer-based approaches can be used to design structured residual sets or fixed residual vectors. For sensor faults, the design of a structured residual set is very straightforward. If we require that a residual is sensitive to faults in all but one of the sensors, the observer used to generate this residual should be driven by outputs excluding that single sensor measurement. To be more specific, if we replace the output
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vector y = (Yl,'" ",Ym) by (yl,.. ",Yi-~,Yi+~,'" ",Ym), the residual will be insensitive to the fault in the ith sensor. However, the design of a structured residual set for actuator fault isolation is more difficult. This problem can be solved via unknown input observers [6,38,39] and eigenstructure assignment [40-42], however the isolation of actuator faults is not always possible. The schemes used in designing observer-based structured residual set have being called the dedicated observer scheme and the generalized observer scheme [5]. The fixed residual vector can be designed by the so-called "fault (or failure) detection filter" originated by Beard [28-30]. B. P A R I T Y R E L A T I O N ( E Q U A T I O N ) A P P R O A C H E S Parity relations (equations) are based either on a technique of direct redundancy, making use of the static algebraic relations between sensor and actuator signals, or alternatively based upon temporal redundancy, when dynamic relations between inputs and outputs are used. The term "parity" was first used in connection with digital logic systems and computer software reliability to enable "parity checks" to be performed for error checking. In the FDI field, it has similar meaning in the context of providing an indicator for the presence of a fault (or error) in system components. In the early developments, parity relation methods were applied to parallel redundancy schemes [43-45]. For such system configurations, the number of measurements is greater than the number of variables to be sensed and the residuals can be obtained directly from the redundant measurements. Inconsistency in the measurement data is then a metric that can be used initially for detecting faults and, subsequently for fault diagnosis. The parity vector concept was generalized by Chow & Willsky [10] and Lou et al [19] for the case of using the temporal redundancy relations of the dynamic system. The redundancy is constructed by collecting sensor outputs over a time interval (data window). Here, we consider that the system is given by the linear discrete state space equations as follows:
x(k+ 1) y(k)
-
-
A x ( k ) + B u ( k ) + Rlf(k) Cx(k) + Du(k) + R2f(k)
(26)
where z E IR~ is state, y E IRm is output, u E IRr is input, f E IRg fault, and A, B, C, D, R1 and R2 are real matrices of compatible dimensions. The redundancy relations are now specified mathematically as follows. Combining together Eq.(26) from time instant k - s to time instant k yields
ROBUST FDI SYSTEMS
185
the following redundant relations:
f(ky(k - s + 1)
u(k - s + 1) ~
H
f(k. - s + 1)
Wx(k-s)+M
-
9
o
9
o
/(k)
u(k) '~
j
J
%,
Y(k)
_J
%,
F(k)
u(k)
(27)
where D CB H
_
0 D
... .
0 ~ o
O
9
_
9
9 9
~
CAS-1B
CAS-2B
9o o
E IR (~+1)'~x(~+1)"
o D
C CA W-
.
E IR ( s + l ) ' ~ x ' ~
CA ~
and the matrix M is constructed by replacing { D , B } with {R2, R1} in the matrix H. To simplify the notation, Eq.(27) can be rewritten as: Y(k) - HU(k) - Wx(k-
s) + M F ( k )
(28)
According to [10,19], a residual signal can be defined as" r(k) - V [ Y ( k ) -
HU(k)]
(29)
where V E IRp• and p is the residual vector dimension. Eq.(29) is termed an Sth order parity equation or parity relation. It is the computational form of a residual generator which shows the residual signal as a function of measured inputs and outputs of the monitored system. Substituting Eq. (28) into Eq.(29), we have: r(k) - V W x ( k -
s)+ VMF(k)
(30)
This is the evaluation format of the residual. In order to make the parity vector useful for FDI, one should make it insensitive to system inputs and states, i.e. VW
-
0
(31)
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To satisfy the fault detectability condition, the matrix V should also satisfy the following condition: VM r 0 (32) Once we have the matrix V, the residual signal can be generated using Eq. (29). Here, we discussed the construction of parity relations based a state space model which is suggested in [10]. However, it must be pointed out that the parity relation can also be constructed using a z-transformed inputoutput model (or discrete transfer matrix representation). Gertler et al [46,47] first introduced this design approach, preferring to call it the "parity equation" method. Gertler [11] presented an excellent tutorial on this approach. Note that this input-output model based design method has also been studied in [33]. The parity relation approach can be used to design structured residual set for fault isolation [7,11,46-48]. The design for isolating sensor faults is very straightforward. If we use cT (the ith row of C) and yi (the ith component of y) instead of C and y, the parity relation will only contain the ith sensor's output together with all inputs. This form of parity relation has been called a single-sensor parity relation [48] and the residual generated by this relation is only sensitive to the fault in the ith sensor. For the actuator isolation problem, the structured residual set is more difficult to design. Massoumnia et al [48] studied this problem and pointed out that the isolation of actuator faults is not always possible. This conclusion is consistent with the observer-based approaches. Gertler et al [11,46,47,49] suggested a so-called "orthogonal parity equation" approach in designing structured residual sets. The idea is to make the parity equation (and residual) orthogonal to a particular fault direction if we want the residual insensitive to this fault. It is not easy to design directional residual vector using parity relations. Gertler [11] studied this problem and illustrated the possibility based on examples, however a systematic approach still does not exist. C. T H E R E L A T I O N S H I P PARITY RELATIONS
BETWEEN
OBSERVER
FDI AND
An interesting feature of parity space methods for FDI is that, given certain design conditions, the parity equation residual signals can become identical to those resulting from a discrete-time observer. This property was noted by Massoumnia [50] and Wiinnenberg [27]. A full derivation of this equivalence has been given by Patton &Chen [7,51]. Essentially, when the discrete-time observer has a dead-beat structure (with all eigenvalues zero-valued), the observer residual is identical to the parity space structure.
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Considering the generalized observer FDI problem, Wiinnenberg & Frank [27,52] give a solution for all matrices in the design of a residual generator and the associated generator matrix. Patton and Chen [7,53] referred to the usual parity space derivation as "an open loop strategy" for residual generation in contrast to a "closed-loop strategy" implied when using the observer approaches. The motivation behind this terminology is simply based on the fact that the observer approaches use a closed-loop feedback construction to yield the required residual, whereas the usual parity space equations do not require the use of any form of feedback. It is also important to note that, under the strict equivalence defined above, the state vector of the system does not enter into the calculation of the residual. It is quite natural, therefore to wonder if indeed it is necessary to make use of methods based on the state space. The final choice of the state space (and hence observers) for FDI, for many studies, seems to be based upon the approach to solving the robustness problem and not requiring explicit use of the state space information. However, when considering non-linear systems in differential equation form, the state-space is essential. The structure of any observer used for FDI purposes can be shown to be equivalent to a parity space scheme cascaded by a suitable transfer function [7,51]. The equivalence between the parity space and observers has also been drawn out by Gertler [11] who also compares the unknown input observer and the parity space with the use of eigenstructure assignment, on the basis of computational cost and complexity. A complete comparison should, of course, involve a study of the relevant advantages/disadvantages of each method in terms of robustness. Such a thorough study is still awaited. D. F A C T O R I S A T I O N TION
METHODS
FOR RESIDUAL
GENERA-
A residual generator can also be synthesized in the frequency domain via factorisation of the transfer function matrix of the monitored system. This method was initiated by Viswanadham et al [34]. A more comprehensive study and extension was made by Ding & Frank [35], in which the FDI problem was systematically formulated and solved via factorisation techniques. This approach is also studied by other investigators such as Marquez & Diduch [54] and Yao et al [55]. The most recent developments including robustness issues can be found in [20]. This approach is based on a fact that any m • r proper rational transfer function matrix G~(s) can be factorized as [56]:
G~(s) = M - l ( s ) N ( s )
(33)
where M(s) and N(s) are rational and realizable transfer function matrices.
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Based on this factorisation, a residual generator can be designed as: r(s) = Q(s)[M(s)y(s) - N(s)u(s)]
(34)
where Q(s) denotes a dynamic residual weighting. On substituting Eq.(2) into Eq.(34) together with Eq.(33), the residual is: r(s) = Q(s)M(s)G/(s)f(s) (35) which is only affected by the fault. The weighting factor Q(s) carl be used to improve the residual performance responding to faults in a particular frequency region. Note that Eq.(34) is a special representation of residual generators which can also fitted into the general framework given by Eq.(5) and Fig.3. The design of a residual generator is to construct the transfer function matrices M(s) and W(s) which can be given by [56]: M(s)
-
N(s)
-
- C [ s I - ( A - KC)]-IL+ I C [ s I - ( A - K C ) ] - I ( B - KD) + D
(36)
On comparing the above equations with the transfer function matrices for a full-order observer-based residual generator given in Eq.(25), one can see that they are almost identical. This demonstrates the correspondence between observer-based and factorisation approaches. E. T H E P A R A M E T E R
ESTIMATION
APPROACH
The parameter estimation approach [2,36] make use of the fact that component faults of a dynamic system can be thought of as reflected in the physical parameters as, for example, friction, mass, viscosity, resistance, inductance, capacitance. This approach is used to detect faults via the estinlation or identification of model parameters using a nonparametric model. F. F A U L T D I A G N O S I S
FOR STOCHASTIC
SYSTEMS
For stochastic systems, the FDI is based on statistical testing of the residuals [1,4,31,32,57,58], for example: 9 The weighted sum-squared residual (WSSR) testing [1,32]. 9 X2 testing [1,57]. 9 Sequential probability ratio testing (SPRT) and modified SPRT [1,31]. 9 Generalized likelihood ratio (GLR) testing [1,31].
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In order to suppress the effect of noise on the residuals, the residual generator has to be specifically designed to deal with the noise. A common approach is the use of Kalman filter-based residual generators. Whilst using a similar structure to the observer, approaches based on the Kalman filter comprise a residual generation mechanism derived by means of a stochastic model of the dynamic system. In normal operation the Kalman filter residual (or innovation)vector (the difference between the measurements and their Kalman filter estimates), is a zero-mean white noise process with known covariance matrix. Mehra & Peschon [31] proposed the use of different statistical tests on the innovation to detect faults in the system. Many variants of the idea of hypothesis testing for FDI [32] have been published since. The idea which is common to all these approaches is to test, amongst all possible hypotheses, that the system has a fault or is fault-free. As each fault type has its own signature, a set of hypotheses can be used and checked for the likelihood that a particular fault has occurred. Some Kalman filter-like state estimators are developed especially for FDI of stochastic systems, e.g: 9 Multiple model adaptive filters (MMAFs) [59]. 9 Two-stage bias-correction filters [60,61].
IV. T H E N E E D F O R R O B U S T N E S S
IN FDI
The reliability of fault diagnosis must be higher than the monitored system. The model-based fault diagnosis is based upon the use of mathematical models of the supervised system. The better the model used to represent the dynamic behaviour of the system, the better will be the chance of improving the reliability and performance in diagnosing faults. However, modelling errors and disturbances in complex engineering systems are inevitable, and hence there is a need to develop robust fault diagnosis algorithms. The robustness of a fault diagnosis system means that it must be only sensitive to faults, even in the presence of model-reality differences (e.g. parameter variations, turbulence, and the effects of manoeuvres) [5,14]. Usually, parameter variations and disturbances act upon a real process in an uncertain way, so that it may be difficult to design a fault diagnosis system which is highly sensitive to faults, whilst insensitive to uncertainty and unmodelled disturbances. The heart of model-based fault diagnosis is the generation of residuals. Both faults and uncertainty affect the residual, and discrimination between their effects is difficult. The task in the design of a robust FDI system is thus to generate residuals which are insensitive to uncertainty, whilst
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at the same time sensitive to faults, and therefore robust [5,13-16]. The robustness is of course only proved if the residual of interest remains insensitive to uncertainty over the whole range of operation of the system being monitored. To approach the problem from the general point view, one nmst start with a mathematical description of the monitored system that includes all kinds of modelling uncertainty that can occur in practice and affect the behaviour of the system. Therefore, the state space model of the system is given by: { x(t)
y(t)
-
(A + AA)x(t)+ (B + AB)u(t)+ Eld(t)+ Rlf(t) (C + AC)x(t) + (D + AD)u(t) + E2d(t) + R2f(t)
(37)
here d(t) E IRq is an unknown input (disturbance) vector, however the unknown input distribution matrices E1 and E2 are assumed to be known. The matrices AA, AB, AC and AD are the parameter errors or variations which represent the modelling errors. The transfer function description of the system is then:
y(s) = (G~(s) + AG~(s))u(s) + Gd(s)d(s) + G/(s)f(s)
(38)
Here Gd(s)d(s) represent the disturbance effect and Gd(s) = E2 + C ( s I A) -1 El, AG~(s) is used to described modelling errors. The terms Gd(s)d(s) and AG~(s)u(s) together represent modelling uncertainty. If we substitute the system output y(s) into the residual generator Eq.(5), the s-domain residual vector is:
r(s) = H~(s)G/(s)f(s) + Hy(s)AG~(s) + Hy(s)Gd(s)d(s)
(39)
Both faults and modelling uncertainty (disturbance and modelling error) affect the residual, and hence discrimination between these two effects is difficult. This is the heart of the robustness problem in FDI [5,13-16]. A. R O B U S T N E S S
TO D I S T U R B A N C E S
If the residual generator has been designed to satisfy:
Hy(s)Gd(s) = 0
(40)
i.e., the disturbance is totally de-coupled from the residual r(t), the residual is robust to the disturbance. This is the principle of disturbance de-coupling for robust residual generation [17,21,40-42,52,62-66]. Disturbance de-coupling designs can be achieved by using the unknown input observer (UIO) [17,21,27,52,62-64] or alternatively, eigenstructure assignment approaches [21,40-42,65,66] These two approaches are studied in
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greater detail in next section. As far as the design of robust residuals is concerned, these methods are formally equivalent whilst using different mathematical tools to achieve the same goal in robustness [11]. Gertler et al [11,47,49] proposed the disturbance de-coupling design based on the so-called orthogonal parity equations. Disturbance de-coupling can also be achieved using frequency domain design techniques (e.g H ~ - n o r m optimization) [20,67-69]. If the condition (40) does not hold, perfect (accurate) de-coupling is not achievable. One can consider an optimal or approximate de-coupling by minimizing a performance index containing a measure of the effects of both disturbances and faults. One suitable choice of performance index can be defined in the frequency domain as [20,68-71]:
j-
liHy(jw)Gd(jw)II IIHy(jw)G/ (Jw)ll
(41)
By minimizing the performance index J over a specified frequency range, an approximate de-coupling design can be achieved [20,68-71]. The optimal approximate disturbance de-coupling design can also be defined and solved in the time domain [27,52]. B. R O B U S T N E S S
TO MODELLING
ERRORS
For modelling errors represented by AG~,(s), the robust problem is more difficult to solve. Two main approaches have been proposed. One, based on an attempt to account for uncertainty in designing the residual is known as active robustness in FDI [14,15,21,49,65,66,71,72] (see Section VI). The second approach is called passive robustness in FDI [13,20,22,68,29], which makes use of adaptive threshold at the decision-making stage and this is discussed in Section VIII. The active way of achieving a robust solution is to obtain an approximate structure for the uncertainty, i.e. to represent approximately modelling errors as disturbances [14,15,21,49,65,66,72,72]: (42) where dl(s) is an unknown vector and Gdl (s) is a estimated transfer function matrix. When this approximate structure is used to design disturbance de-coupling residual generators, a suitably robust FDI is achievable. As the attempt is made to render the actual residuals robust with respect to uncertainty, we call this active robustness in FDI.
V. R O B U S T
FDI FOR DISTURBANCES
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R.J. PATTON AND J. CHEN
A. S Y S T E M
DESCRIPTION
One of the most successful robust fault diagnosis approaches is the use of the disturbance de-coupling principle, in which the residual is designed to be insensitive to unknown disturbances, and meanwhile sensitive to faults. Using the disturbance de-coupling approach, the robust FDI with respect to unknown inputs (disturbances) can be solved. This approach deals with the observer design for a class of systems, in which the system uncertainty can be summarized as an additive unknown disturbance term in the dynamic equation described as follows:
( x(t) y(t)
-
Ax(t) + Bu(t) + Rlf(t) + Ed(t) Cx(t) + R2f(t)
(43)
where x(t) E IR'~ is the state vector, y(t) C ]Rm is the output vector, u(t) E IW is the known input vector and d(t) ~ IRq is the unknown input (or disturbance) vector. A, B, C and E are known matrices with appropriate dimensions. Remarks: (a) There is no loss of generality in assuming that the unknown input distribution matrix E should be full column rank. When this is not the case, the following rank decomposition can be applied to the matrix E:
Ed(t) = Ei E2d(t) where E1 is a full column rank matrix and E2d(t) can now be considered as a new unknown input. (b) The term Ed(t) can be used to describe an additive disturbance as well as a number of other different kinds of modelling unce:rtainties. Examples are: noise, interconnecting terms in large scale systems, nonlinear terms in system dynamics, terms arise from tim(~-varying system dynamics, linearization and model reduction errors, parameter variations. A discussion of this problem can be founded in Se,:tion VI. (c) The disturbance term may also appear in the output equation, i.e.,
y(t) = Cx(t) + E~d(t) This case is not considered here because the disturbance terin Eyd(t) in the output equation can be nulled by simply using a transformation of the output signal y(t), i.e.
v (t) = T y(t) = T C (t) + T E d(t) = T
Cx(t)
ROBUST FDI SYSTEMS
193
where TyEy = 0, if one replaces y(t) and C with yE(t) and TyC, the problem will be equivalent to one without output disturbances. (d) For some systems, there is a term relating the control input u(t) in the system output equation, i.e.
y(t) = C x ( t ) + Du(t) As the control input u(t) is known, a new output can be constructed as:
~(t) = y(t) - Du(t) = C x ( t ) If the output y(t) is replaced by ~(t), the problem will be equivalent to the one without the term Du(t). For brevity, the term Du(t) is omitted as this does not affect the generality of the discussion on the observer design. B. R O B U S T OBSERVERS
RESIDUAL
DESIGN
VIA
UNKNOWN
INPUT
To achieve robust FDI, the diagnostic residual must be designed independent of system uncertainty. The goal of the unknown input observer is to force each of the state estimation errors to become independent of the uncertainty. Once, the state estimation error vector is de-coupled from the uncertainty, it then follows that the residual will also be de-coupled from the uncertainty. The design of unknown input observers has been studied for many years, and the recent developments can be found in [74,75]. In this chapter, a full-order unknown input observer suggested by Chen et al [76] is used. The structure for this full-order observer is described as: /(t)
-
Fx(t)+TBu(t)+Ky(t) z(t) + H
(t)
(44)
where 2 6 IR'~ is the estimated state vector and z 6 IR'~ is the state of this full-order observer, and F, T, K, H are matrices to be designed for achieving unknown input de-coupling and other design requirements. The residual is defined as:
r(t) = y(t) - C2(t) = (I - C H ) y ( t ) - Cz(t)
(45)
If one can make the following relations hold true: (HC-
I)E
-
T
-
0 I-HC
F
-
A-HCA-KIC
K2
-
FH
K
-
K1 + K 2
(46)
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R.J. PATTON AND J. CHEN
The estimation error (e(t) = x ( t ) - &(t)) and residual are governed by the equation:
~(t) r(t)
=
Fe(t) + T B R l f ( t ) Ce(t) + R2f(t)
-
I~-'lR2f(t)
-
HR2:(t).
(47) (48)
If all eigenvalues of F are stable, e(t) will approach zero asymptotically if there is no fault in the system, i.e. the residual is robust against to disturbance d(t). The design of this robust residual generator is to solve Eq.(46) and making all eigenvalues of the system matrix F be stable. This can be achieved using the following theorem: T h e o r e m 1: Necessary and sufficient conditions for (44) is a robust residual generator for the system defined by (43) are: (i) r a n k ( C E ) :
rank(E)
(ii) (C, A1) is detectable pair, where
A1 - A - E[(CE) T CE]-I(CE) T CA
(49)
One should note that the number of independent row of the matri~ C must not be less than the number of the independent columns of the matrix E to satisfy condition (i). That is to say, the maximum number of disturbances which can be de-coupled cannot be larger than the number of the independent measurements. It is very interesting to note that observer (44) will be a simple full-order Luenberger observer by setting T = I and H == 0, when E = 0 (i.e. no unknown inputs in the system). In this situation, condition (i) in Theorem 1 is clearly hold true and condition (ii) is simply changed to that of (C, A) being detectable. This is a well known result in the design of a full-order Luenberger observer. Condition (ii) can be verified in terms of the structural properties of the original system. In fact, this condition is equivalent to the condition that the transmission zeros from the unknown inputs to the measurements must be stable, i.e.
[sI~-A
0]
E
is of full column rank for all s with Re(s) >_ 0 [76]. C. T H E E I G E N S T R U C T U R E
ASSIGNMENT
APPROACH
The unknown input observer is a robust observer in which FDI residuals become insensitive to uncertainty. For FDI purposes, it is not necessary to ensure that the state estimates have this insensitivity. Following this line
ROBUST FDI SYSTEMS
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of reasoning Patton and co-workers [21,40-42,65,66] have studied the use of eigenstructure assignment for linear observer design, with the FDI problem as the principle goal. Eigenstructure assignment is a very powerful approach to the design of any linear state-space feedback system, as it can be shown very easily that the closed-loop system structure depends entirely upon (a) the system eigenvalues and (b) the system's left and right eigenvectors. In the eigenstructure assignment approach, the residual is generated using an observer with the following structure: { ~(t)
-
Ac~(t) +Bu(t) + Ky(t)
9(t)
-
c
(50)
(t)
where d~(t) E IR~ is the state estimate vector and ~)(t) E IRm is the output estimation vector and Ac = A - KC. In this case the state estimation error vector is e(t) = x ( t ) - 2(t). The error dynamics are:
~(t) = Ace(t) + Ed(t) + Rlf(t) - KR2f(t)
(51)
The residual vector r(t) C lRp is defined as: =
=
Q(y(t)
-
y(t))
(52)
where Q is a suitable weighting matrix designed to give good isolability properties and insensitivity to uncertainty. r(t) = WCe(t) + QR2f(t)
(53)
From Eqs. (51) & (53), the complete response of the residual vector is: r(s) - QR2f(s) + QC(sI - Ac)-I(R1 - KR2)f(s) + QV(sI - Ac)-lEd(s) (54) Clearly, the residual responds to both faults and uncertainty. In order to minimize the false alarm rate resulting from the uncertainty, the residuals must be de-coupled from the disturbances and uncertainty. The de-coupling conditions are hence equivalent to requiring that entries in the transfer function matrix between disturbances and the residual be nulled, i.e. Q C [ s I - ( A - KC)]-IE = 0 (55) Patton [46] noted that this problem is a special case of the output-zeroing problem which is well known in multivariable control theory [77]. Alternatively, the residual vector will have zero-sensitivity to uncertainty when Eq.(55) is satisfied. The remaining problem is to choose the matrices K and Q to satisfy Eq.(55), in addition to choosing eigenvalues to optimize
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R.J. PATTON AND J. CHEN
the FDI performance. As the observer (and hence the residual) is a linear time-invariant multivariable system, it can be uniquely defined from a feedback point of view using the observer's eigenstructure. The assignment of the observer's eigenvectors and eigenvalues is a direct way to solve the required FDI design problem. Two Lemmas show how the relationship between the eigenvectors and eigenvalues (the eigenstructure) of a system can be used as follows: L e m m a 1" A given left eigenvector lTi (corresponding to the eigenvalue /3/) of Ac is always orthogonal 1;o the right eigenvectors vj corresponding to the remaining (n-l) eigenvalues ~j of Ac, where t3/7 ~/3j. L e m m a 2: The resolvent matrix can be expanded in terms of the eigenstructure as: ( s I - A ~ ) _ 1 = vl lT + . . . + (56) where vi and l/T are the right and left eigenvectors of Ac respectively, corresponding to the eigenvalue ]3/. T h e o r e m 2: If Q C E = 0 and all rows of the matrix H = QC are left eigenvectors of Ar corresponding to p eigenvalues of Ar then Eq.(55) is satisfied. Proof: Let all rows of the matrix H - QC are p left eigenvectors (1T, i 1, 2 , . . . , p ) of Ac, i.e. g -
ly then: H v i = 0 for i = p + 1 , . . . , n, and
H(sI-
Ac)-IE-
H
-t- ""-t- s _ ~p
1
E1
But Q C E - H E - O, i.e., 1Ti E -- O, for i -- 1 , 2 , . . . , p . Thus: Grd(S) = Q C ( s I - Ar = 0.
QED
When the residual is completely de-coupled from the disturbances (unknown inputs), robust FDI is achievable. The eigenstructure assignment approach can be summarized as: (1) Compute the residual weighting matrix Q so that Q C E = O. (2) Determine the eigenstructure of the observer: The observer elgenvalues are chosen according to the desired dynamic property of residuals. The rows of H = QC must be p left eigenvectors of the observer, the remaining ( n - p) eigenvectors can be chosen freely.
ROBUST FDI SYSTEMS
197
(3) Compute the gain matrix K: Assign the required eigenstructure of the observer using an assignment algorithm. The observer eigenstructure assignment problem can be handled by means of a transformation of the dual control form. On assignment of the right eigenvectors to the dual control problem these eigenvectors become the left eigenvectors of the observer system [47]. The assignability condition is that, for each eigenvalue /3i, the corresponding left eigenvector l_T must belong to the row subspace spanned by: { C ( / 3 i I - A)-I}. If the left eigenvector assignability condition is not satisfied, a suitable choice of right eigenvectors can be assigned as columns of the matrix E. T h e o r e m 3: If Q C E - 0 and all columns of the matrix E are right eigenvectors of Ac corresponding to any eigenvalues, Eq.(55) is satisfied [21,42]. If the desired observer eigenstructure is assignable (using the left or right eigenvectors), perfect de-coupling can be achieved. On the other hand, if the required eigenstructure is not perfectly assignable, the eigenvectors must be chosen to be close in a least-squares sense, to the desired eigenvectors. In this situation, the residuals also have low sensitivity to uncertainties due to approximate de-coupling. Hence, for a given matrix E, the disturbance de-coupling problem can be solved iff a matrix H can be found which requires the following rank condition: rank(E) < n - p (57) As far as the design of robust residuals is concerned, the unknown input observer and the eigenstructure assignment FDI observer are formally equivalent and use different mathematical tools to achieve the same goal in robustness [11]. However, the use of eigenstructure design for an identity observer employs fewer design constraints than the unknown input observer approach. The unused design freedom can then be used for disturbance de-coupling and or fault isolation. E. R O B U S T
FAULT ISOLATION
SCHEMES
The fault isolation problem is to locate the fault, i.e., to determine in which sensor (or actuator) the fault has occurred. In robust fault isolation, diagnostic residuals have to be designed have robust as well as isolable properties. Hence, the design problem becomes difficult. As pointed out in Section II, one of the approaches to facilitate fault isolation is to design a structured residual set. The term "structured" here means that each residual is designed to be sensitive to a certain group of faults and insensitive to others. The sensitivity and insensitivity property makes isolation possible. In
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R.J. PATTON AND J. C H E N
robust fault isolation, a robust structured residual set should be designed. The design consists two stages, the first one is to specify isolable structure, and the second is to make each residual robust. To exploit the maximum design freedom for robustness, a commonly accepted scheme [5,21] in fault isolation is to make each residual to be sensitive to faults in all but one sensors (or actuators). Robust isolation can also be achieved using robust directional residuals. There have been some studies on this problem, for example, Olin & Rizzoni [78] tried to make fault detection filter robust using the time-frequency distribution concept, Chen et al [76] combined unknown input observer and fault detection filter principles to design robust directional residuals. However, the robust isolation using robust directional residuals is still a problem for further research. In the following, the robust fault isolation schemes based on structured residual sets are presented. Note that, each robust residual is designed via an unknown input observer. However, the robust isolation can also be achieved using the eigenstructure assignment approach or the orothogonal parity relations approach. Robust sensor fault isolation schemes To design robust sensor fault isolation schemes, all actuators are assumed to be fault-free and the system equations can be expressed as: ,
,,
5:(t)
yJ (t) yj(t)
--
A x ( t ) + Bu(t) + Ed(t) C j x(t) + g (t) cjx(t) + fsj(t)
for
j-
1,...,m
(58)
where cj E R l• is the jth row of the matrix C, C j E IR (m-1)• is obtained from the matrix C by deleting jth row ci, yj is the jth component of y and yJ E IRm-1 is obtained from the vector y by deleting jth component yj. Based on this description, m UIO-based residual generator can be constructed as:
{ iJ (t) (t)
-
FJ zJ (t) + TJ Bu(t) + KJ y.i (t)
-
(I -
(t) -
CJ
(t)
for
J
- 1 ' '
'
Tn
(59)
where the parameter matrices must satisfy the following equations: HJc j E Tj
-
E I_HJC
FJ KJ2 KJ
-
TJ A - K~ C j FJ HJ K{ + Ii~
j
stable
for j - 1 , . . . , m
(60)
It is clear that each residual generator is driven by all inputs and all but one outputs. When all actuators are fault-free and a fault occurs in the jth
ROBUSTFDI SYSTEMS
199
sensor, the residual will satisfy the following isolation logic:
{ ll,-~(t)ll TJF, <
Ilr k (t)l I >_ T~F I
(61)
for k - l , . . . , j - l , j + l , . . . , m
where T~F I (j -- 1 , . . . , m ) are isolation thresholds. A robust and UIObased sensor fault isolation scheme is shown in Fig.6. input u ? I ACTUATORS ][ ~ PLANT
output ~
SENSORS
I
~ Signal _ I . I Gr~ 1
-~
UIO 1
!
[z ~ C 1
I I I I
9
L~UIO
m
.
+
l+
I I" C1H1 ~'J Y' ' - - - - - ] - ~ rl(t)
[I'CmH
m [- ym
rm(t)
i Zm
Figure 6: A robust sensor fault isolation scheme R o b u s t a c t u a t o r fault isolation schemes To design robust actuator fault isolation schemes, all sensors are assumed to be fault-free and the system equation can be described as: { 2(t)
-
= -
y(t)
A x ( t ) + Biui(t) + B i f i ( t ) + bi(ui(t) + fai(t)) + Ed(t) A x ( t ) + Biui(t) + Bifia(t) + Eidi(t) Cx(t) for i - 1 , . . . , r
(62) where bi E ]R n is the ith column of the matrix B, B i E ]Rn• is obtained from the matrix B by deleting the ith column bi, ui is the ith component of u, u i E IR r-1 is obtained from the vector u by deleting the ith component ui , and E i-
[E
bi]
di(t)-
ui(t) d(t) + fai(t) ]
for
i-1,2,..-r
200
R . J . P A T T O N A N D J. C H E N
Based on the above system description, r UIO-based residual generators can be constructed as: { ii(t) ri(t)
-
for i - 1
rizi(t) + TiBiui(t)+ Kiy(t) (I- CHi)y(t) - Czi(t)
0
0
0
'
'
(63)
r
The parameter matrices must be satisfy the following equations: HiCEi Ti Fi Ki2 K i
--
Ei I - Hic T iA- K~C Fill i It'~+Ki2
stable
for i - 1 , . . . , r
(64)
One can see that each residual generator is driven by all outputs and all but one inputs. When all sensors are fault-free and a fault occurs in the ith actuator, the residual will satisfy the following isolation logic:
{ II~' ~(t)ll < II~'k(t)ll _
T~4FI
TkAF~
for
k - 1,..., i-
(65)
1, i + 1 , . . . , r
where TAR i I ( i - - 1,..., r) are isolation thresholds. A robust and UIO-based actuator fault isolation scheme is shown in Fig.7. -
I ACTUATORS
input -
~
PLANT
SENSORS
]yOUtput -
Signal k.
Grouping
Y
I'CH1 .r rl(t)
nile, UIO 1 ~
],j-,
I-
C ~
I I 9
9
Ur
"ld UIOr
~-~
{ I-CH
,k. Y r r"(t)
C ~- ~
Figure 7" A robust actuator fault isolation scheme
.r 2
I
'i ~ ,,-t
ROBUST FDI SYSTEMS
201
R e m a r k s : The isolation schemes can also be designed using eigenstructure assignment method. The isolation schemes presented in this section can only isolate a single fault in either a sensor or an actuator, at the same time. This is based on the fact that the probability for two or more faults to occur at the same time is very small in a real situation. If simultaneous faults need to be isolated, the fault isolation scheme should be modified via a regrouping of faults. Each residual will be designed to be sensitive to one group of faults and insensitive to another group of faults. The way of grouping faults is dependent on the system and the faults to be isolated. The isolation of sensor faults is normally possible, however it is impossible to isolate two actuator faults which have the same distribution direction. To isolate such actuator faults, other fault information such as fault frequency distributions should be utilised. FDI schemes are related to particular systems, a general scheme cannot expected to suit any system without any modification.
VI. R O B U S T
FDI FOR MODELLING
ERRORS
In order to design robust FDI schemes, we need the description of uncertainties acting upon the system, during typical process plant operations. Furthermore, it is necessary to find a description of these uncertainties which can be handled in a straightforward and systematic manner. Whilst all system uncertainties can be summarised as unknown inputs (disturbances) acting on the system, their effect can be considered as bounded or unbounded and structured or unstructured. When an a t t e m p t is made to make the residuals robust with respect to uncertainty, we call this active robustness in FDI [14,15]. When, on the other hand, the robustness is focused on the decision-making stage of FDI, this is termed a passive approach to robust FDI, for example using adaptive threshold techniques (see Section VIII). The most powerful approaches to robust FDI make direct use of disturbance de-coupling, according to the most readily available estimates of the uncertainty, given an operating point of interest. In particular, approaches making use of the special design of observers are important. An observer used for fault diagnosis can be designed to be robust in the sense of disturbance de-coupling; the robustness will ensure t h a t the observer residual is insensitive to disturbances and modelling errors (Section V). For all disturbance de-coupling methods, an important assumption is that the unknown input distribution matrix E must be known, however their derivation is, apart from very simple examples, a very complex problem. As discussed in Section IV. B., one of the ways to tackle this problem is to represent modelling errors as a disturbance term with an estimated
202
R.J. PATTONAND J. CHEN
distribution matrix. This estimated distribution matrix is then used to design disturbance de-coupling residuals, in this way, optimal robust FDI solutions are achievable. This idea was originally proposed by Patton and Chen [65,66] and now has been widely accepted in robust FDI research [14,15,21,49,72,73]. In most situations, the distribution matrix is not readily available. However, there are cases for which some a priori knowledge about uncertainty is available and can be used for a direct derivation of the distribution matrix E. This Section discusses a number of situations in which some realistic assumptions about uncertainty can be used for this direct derivation. Normally, this directly obtained matrix has a high rank (i.e. too many disturbances or unknown inputs) and disturbance de-coupling is not achievable. Hence, a low rank matrix which approximates the distribution is used in the design of optimally robust residual generators. This is an unknown input consolidation procedure, i.e., the unknown inputs with closed directions are combined and hence the number of unknown inputs is reduced. Note that, in some situations, the matrix E can be determined by simple inspection. If the uncertain factors appear in the ith row of matrices A and B, it is most likely the matrix E should contain a column as follows: [
0
...
0
1
0
...
0
Ir
ith This direct inspection method for determining the matrix E was also discussed in [21,73,75,79]. This method may not be very effective, however it is simple and can be useful for some systems. A. N O I S E A N D
ADDITIVE
NON-LINEARITY
Consider the following dynamic equation of the monitored system:
2(t) = Ax(t) + Bu(t) + Gp(t) + r f(x(t), u(t), t))
(66)
where p(t) is a noise or external disturbance vector. In this equation, the non-linearity is considered as an additive non-linear term Qf(x(t), u(t), t)), i.e., the system dynamics can be separated into linear and non-linear parts. This kind of non-linear dynamic structure exists in some non-linear chemical processes [17,21,64]. For the system described above, the uncertainty can be modelled as an additive term Ed(t) and where:
Ed(t) - [G F] B. B I L I N E A R
SYSTEMS
[ f(x(t t'/u(t), t)) ]
(67)
ROBUST FDI SYSTEMS
203
The study of bilinear systems has theoretical importance because they are a special class of nonlinear systems. Many practical nonlinear systems such as ecological systems, nuclear systems, hydraulic systems and heat exchanger systems can be modelled by a bilinear system model [24]:
x(t) - Aox(t) + Bu(t) + f i Aiui(t)x(t)
(68)
i--1
where ui(t) (i = 1,...,r) is the ith component of u(t), and Ai (i = 0, 1,.-., r) and B are known matrices. The nonlinear term can be treated as the disturbance term with the distribution matrix and the unknown input vector as follows:
I ul(t)x(t) E-[A1
A2 "." At]
d(t)-
:
(69)
A linear disturbance de-coupled residual generator can be designed to generate robust residuals for FDI. This avoids the complexity involved in the design of bilinear observers. C. M O D E L
REDUCTION
Most systems can have significantly higher order dynamics than their models. Consider, for example, the system described by a higher order model as [41,66]"
-
+
u(t)
(70)
where x(t) E IR'~ is a partial state vector corresponding to dominant dynamic part of the system, xh(t) represents the higher order dynamics in the system, and frequently neglected in practice. For ease of design and implementation in control and fault diagnosis, the following reduced-order model is used to approximate this system"
ie(t) - Ax(t) + Bu(t) + (111 - A)x(t) + (B1 - B)u(t) + A12xh(t) = Ax(t) + Bu(t) + Ed(t) (71) where"
x(t) ] Ed(t) - [(All - A )
(B1 - B) A12] u(t)
h(t)
(72)
A typical application of this partitioned state-space structure arises when comparing a reduced order model with the full-scale system, for example,
204
R . J . PATTON A N D J. C H E N
in an observer used for FDI. For this case, the nominal model represented by (A, B) is the reduced order model and the remaining modellirlg errors are considered to be lumped together within an additive term s It is assumed that the n reduced order state variables correspond to N state variables of the full-scale system. D. P A R A M E T E R
PERTURBATIONS
A system model with time-varying parameter perturbation can be described as"
x(t) - (A + AA(t))x(t) + (B + AB(t))u(t)
(73)
The parameter perturbations considered in the robust control field are sometimes approximated as: N
AA(t) ~ E
N
ai(t)A,
bi(t)Bi
AB(t) ~ E
i--1
i--1
where Ai and Bi are known constant matrices with proper dimensions, ai(t) and bi(t) are unknown scalar time-varying factors. In this ,:ase, the modelling error can be approximated by the disturbance term as"
al(t)x(t) 9
E d ( t ) - AA(t)x(t) + AB(t)u(t) - [A1 "" AN B1 "" BN]
.
.
ag(t)x(t) bl(t)u(t) 9
.
,
bg(t)u(t) Now, consider the situation where the system matrices are functions of the parameter vector a E IRg"
x(t) - A(a)x(t) + B(a)u(t)
(74)
If the parameter vector is perturbed around the nominal value c~ -- a0, this equation can be expanded as: g OA it(t) - A(ao)x(t) + B(ao)u(t) + E{~Saixoai
OB + ~Saiui[
(75)
i--1
In this case, the distribution matrix and unknown input vector are:
OA
OB
OA
OB
(76)
ROBUST FDI SYSTEMS
d ( t ) - [6 1 x T I E. L O W R A N K TRIX
uT
I... I
APPROXIMATION
205
x T
I
T
OF DISTRIBUTION
(77) MA-
The key principle behind achieving robustness (disturbance de-coupling) is to find a p x n matrix H to satisfy Eq.(55). If r a n k ( E ) <_ n - p , Eq.(55) has solutions and exact de-coupling is possible. If, however, r a n k ( E ) > n " p, Eq.(55) has no solutions and exact de-coupling is impossible. Here, we consider the more general problem in which approximate de-coupling is necessary. The procedure will be to first compute a matrix E* that is as close as possible to E, and r a n k ( E * ) <_ n - p , i.e. to determine the matrices E* and H so that: HE* = 0
with
liE - E*[[~
(78) minimized
(79)
here II.ll denotes the Frobenius matrix. The optimization constraint condition (78) is equivalent to requiring that r a n k ( E * ) <_ n - p. If only p linearly independent rows of H are required, the constraint condition can be changed to r a n k ( E * ) = n - p , so that E* is chosen to minimize (subject to the constraint that E* contains only n - p linearly independent columns) the sum of squared distances between the columns of E and E*. This optimization problem is easy to solve using Singular value Decomposition (SVD) of E as follows [19,26]: E = Z [ d i a g { ( 7 1 , . . . , crn},
0](I)
(80)
where Z and (I) are orthogonal matrices, crl _ (r2 _ ... < cr~ are the Singular Values of E. As shown in Lou et al [19], the matrix E* which minimizes (79) is given by: E* = Z[diag{O, . . ., O, Crp+l,..., (r~},
0](I)
(81)
and an orthonormal solution of the matrix H is: H = [zl z2 ".. zp] T
(82)
where z l , . . . , zp are first p columns of S. Certainly, the matrix H in (82) and E* in (81) together satisfy the Eq. (78). An alternative statement of the optimization problem is given by Patton & Chen [66]. An ideal matrix H should make H e i = 0 for all i = 1, 2 , . . - , nl, where ei is the ith column of the matrix E. This is not always possible.
206
R . J . P A T T O N A N D J. C H E N
Hence, it makes sense is to choose a matrix H that is "as orthogonal as possible" to all ei(i - 1 , 2 , . . . , n l ) , i.e. to make each ofHei(i - 1, 2 , . . . , n l ) as close to zero as possible. The optimization criterion can be defined as: J - ~ i ~ 1 IIHeill2F 9 The optimal solution for H follows by minimizing J, subject to H H T - I. Lou et al [19] show that the choice of H given in (82) also minimizes J, yielding the minimum value as J - ~-'~i=1 P o'2. '['he new statement of the optimization problem provides some very useful insight as J* can be used as a robustness measure which is clearly relative to the independent row number p of the matrix H. Typically, some components of the unknown input vector d are larger than others. Furthermore, certain components of the unknown input vector have more effect on the residual. Hence we must pay varied attention to the different components of the disturbance in the optimization procedure [66]. For example, if the jth component of the disturbance is significantly larger than the i - t h component, the term H e j will be more important than nl the term Hei 9 The criterion J must be replaced by: J - ~ i = 1 c~i~lHeil] F2, where ~ i ( i - 1, 2 , . . . , n l ) are positive weighting factors. The relative magnitudes of the c~i correspond to relative magnitudes of components of the disturbance weighting. By rewriting the weighting optimization criterion nl as: J - ~ i = 1 [[g(x/~zeil]2F, the problem can be solved using the procedure described above, but with ei replaced by ~5"[ei and with E replaced by E'--[~el~/~e~ ..- ~XZ:,e~,]. F. B O U N D E D
UNCERTAINTY
Now, consider the case when the full-order system model is not available. An identification procedure is used to obtain the nominal model {A0, B0, Co, D0 } with the estimation error {AA, AB, AC, AD}. Normally, AA and AB are unknown but bounded [66]: A1 <
AA
_
(83)
B1 <
AB
(84)
where A1, A2, BI and B2 are known and AA < A2 denotes that each element of AA is not larger than the corresponding element of A:a. This typifies the case where the uncertainty is bounded. Consider AA and AB in a finite set of possibilities, say {AAi, ABi} (i - 1 , 2 , . . . , M ) within the interval A1 _< AA _< A2 and B1 < AB < B2. This might involve choosing representative points, reflecting desired weighting on the likelihood or importance of particular sets of parameters. In this situation, a set of unknown input distribution matrices is obtained: Ei - [ A A i , A B i ]
i - 1,2,..., M
(85)
ROBUST FDI SYSTEMS
207
In order to make the disturbance de-coupling valid for a wide range of model parameter variations, an optimal matrix E* should be as close as possible to all E i ( i - 1, 2 , . . . , M ) . The optimization problem is thus defined as: min {s.t. rank(E*)<_m}
lIE*-[E~
E2
...
EM]II2F
(86)
E* is then used to design disturbance de-coupling robust residual generators. As E* is close to all Ei, approximate de-coupling is achieved over the whole range of parameter variations. G. O P T I M A L D I S T R I B U T I O N M A T R I X F O R V A R I E D O P E R ATING POINTS
The operating point of the system varies according to different plant conditions, and different operating points correspond to different unknown input matrices, E i ( i = 1, 2 , . . . , M ) . It is attractive to be able design a single FDI scheme for a whole range (or a set) of operating points [66]. The success of the single FDI design depends on its robustness properties. In order to make the disturbance de-coupling hold for all operating points, we must In ake: H E i = 0, or
HIE1
for E2
""
(87) (ss)
i = 1,2,...,M EM] -- H P = 0
If r a n k ( P ) < n - p, the above equation has solutions and the exact decoupling at all operating points is achievable. If r a n k ( P ) > n - p, approximate de-coupling must be used. This is equivalent to the solution of Eq. (78) and can be solved by defining an optimization problem: min l I P - P*ll~ 3H ~: 0 for g P * = 0
subject to: VII. ROBUST PROACHES
FDI
VIA
FREQUENCY
DOMAIN
(89) (90)
AP-
A. T H E M A R Q U E Z A N D D I D U C H R O B U S T O B S E R V E R
Marquez and Diduch [54] developed a new observer structure for FDI, based on unstructured uncertainty, however their work belongs essentially to a class of frequency domain approaches, based on sensitivity optimization. In an attempt to overcome the limitations on achievable sensitivity imposed by the structure of classical state observers, Marquez and Diduch
208
R.J. PATTON AND J. CHEN
were motivated to find a new generalised observer structure which provides additional degrees of freedom in shaping the observers sensitivity to unknown inputs. They gave a parameterisation of all stabilizing generalized observers, through an observer sensitivity minimisation problem, via Hoo optimization. The starting point of their work was to consider the frequency domain model of the nominal plant (arising from the system of Eq. (1), initially), i.e.:
G~(s) = C ( s I - A ) - I B + D
(91)
The sensitivity function for the identity observer for this system is:
So(s) = [I + C ( s I - A)-IK] -*
(92)
K is the gain matrix of the full order observer. From (92), it is clear that a measure of sensitivity-robustness of the FDI scheme, is the oo-norm of the observer sensitivity IISo(jw)ll, where I1.1]oo= max(or{.})- the maximum singular value of the sensitivity function, over all frequencies w. As Marquez and Diduch point out, there are a number of serious limitations inherent in the classical state space observer structure: (a) The order of the resulting observer sensitivity, S0(s), is limited to the order of the plant, G~,(s). (b) The zeros of So(s) are identical to the poles of the original plant, and (c) It is difficult to relate frequency domain constraints on So(s) with the choice of the observer gain matrix K. Limitation (a) implies small gain at high frequencies, and consequently a small lie[[, may not be achievable when reduced order models are used. Limitation (b) implies that one does not have complete freedom in the frequency domain shaping of the observer sensitivity. Every zero of So(s) will increase the slope of the observer sensitivity by +20dB per decade at high frequencies. Consequently, if G~,(s) does not have high frequency poles, then So(j~z) will inevitably be large at high frequencies. Marquez and Diduch gave a simple example showing that the full order identity observer will actually attenuate low frequency effects. Since steady-state residuals are being used for decision-making, the observer will attenuate the steady-state misbehaviour caused by a fault. The partial state observer considered by Viswanadham and Minto [67] does not improve matters. On considering the dual coprime factorisation [Nr(s), Dr(s)] and [Nz(s), Dr(s)], of the plant (where Ni(s), Dl(s) and Nr(s), Dr(s) are RHoo matrices) referred to in Section III.D., recall that: :
:
(93)
ROBUSTFDI SYSTEMS
209
Marquez and Diduch [54] show that the optimal partial state observer with zero estimation error in the partial state (considering the nominal or unperturbed system), is actually given by Eq.(93) and this corresponds to an open-loop observer (with gain K = 0). The sensitivity for this "optimal" case is given by: So(s) = 1 (94) As this is clearly of no value, when considering the robustness to uncertainty (via frequency-domain sensitivity), Marquez and Diduch then proceeded to develop what they call the general ~observer, which we will now call the "Marquez and Diduch" observer. This observer comprises the nominal plant frequency domain model G~(s) given in Eq.(91), together with a stabilizing generalized observer gain K(s), yielding an observer sensitivity of:
So(s) = [I + C(sI - A) -1BK(s) + OK(s)] -1
(95)
Marquez and Diduch then proceed to show how the norm IISo(jw)ll~ of the above sensitivity function can be made appropriately small at high frequencies, to minimize the effect of modelling uncertainty. To perform this minimisation they propose the use of a weighting function W(jw), such that, for some chosen a:
]lSo(jco)l]~o __ c~
(96)
which reduces to a problem of finding a Q(s) such that:
IIW(jw)[Y~(ja~)- N~(jw)Q(jw)]Dz(jw)l I < a
(97)
such that there exist RHcr matrices Yr(s), Xr(s), Y}(s), Xl(s), satisfying appropriate forward and reverse Bezout identities. This is a standard Hr162 optimization problem. Optimization problems of this kind can, of course, be solved using different approaches. B. M U L T I - O B J E C T I V E
METHODS
A new and systematic approach to the design of robust residuals for detecting incipient faults has been given by Chen et al [21,80,81], which uses multiobjective Optimization. To reduce false and missed alarm rates in fault detection, a number of performance indices are introduced into the observer design. Some performance indices are expressed in the frequency domain to take account of the frequency distribution of faults, noise signals and modelling uncertainties. All objectives are reformulated into a set of inequality constraints on the performance indices. Although the approach taken has been based on the identity observer, and the residual is simply the primary residual or output estimation error
210
R.J. PATTON AND J. CHEN
ey(t), the use of frequency domain optimization becomes and even further generalization of the optimized observer structure by Marquez and Diduch [54]. The essential difference is that, instead of considering the optimization of the estimation error, the residual is optimized over suitable frequency bands. By doing this, the actual structure of the observer used is less important. The use of the residual rather than the state estimation error is, of course, more relevant to the FDI problem. Examining only the case of sensor faults and considering the identity observer FDI problem, the Laplace transform of the residual r(t) is given by:
r(s)
+ C ( s I - Ae)-l[d(s) + e(0)]
-
[I - C ( s I - A r
=
G~l(s, K ) f ( s ) + G~d(S, K)[d(s) + e(O)]
(98)
where e(0) is the initial value of the state estimation error. Both faults and disturbances affect the residual, and discrimination between these two effects is difficult. To reduce false and missed alarm rates, the effect of faults on the residual should be maximized and the effect of disturbances on the residual should be made minimized. One can maximize the effect of the faults by maximizing the following performance index, in the required frequency range [(a.)l,W2]: inf
Z { [ I - C ( j w I - Ar
(99)
This equation is equivalent to the minimisation of the following performance index 9 JI(K) sup N { [ I - C ( j c o I - Ar -1} (100) ~E[~I ,'-"d where s and N{.} denote the minimal and maximal singular values. One can minimize the effects of both disturbance and initial condition by minimizing the following performance index: J2(K) -
sup
- # { C ( j w I - Ac) -1}
(101)
Besides faults and disturbances, the noise in the system can also affect the residual. To reduce the noise effect on the residual, we need to minimize" II I -
c(jwI-
A c ) - I K 11oo
This contradicts the requirement of maximizing the effects of faults on the residual. Fortunately, the frequency ranges of the faults and noise are normally different. For an incipient fault signal, the fault information is contained within a low frequency band as the fault development is slow. However, the noise comprises mainly high frequencies signals. Based on these
ROBUSTFDI SYSTEMS
211
observations, the effects of noise and faults can be separated by using different frequency-dependent weighting penalties. In this case, the performance index J I ( K ) will be:
JI(K) -
sup
-~{Wl(jw)[I - C ( j w I - Ac)-IK] -1}
(102)
To minimize the effect of noise on the residual, a new performance objective J3(K) is introduced as: J3(K) -
sup -~{W3(jw)[I- C ( j w I ~e[wl,w2]
Ac)-IK]}
(103)
In order to maximize the effects of faults at low frequencies and minimize the noise effect at high frequencies, the frequency-dependent weighting factor Wl(s) should have large magnitude in the low frequency range and small magnitude at high frequencies. The frequency effect of W3(s) should be opposite to Wl(s) and can be chosen as W3(s) = W11(s). The disturbance (or modelling error) and input noise affect the residual in the same way. As both effects should be minimized, the performance index J2 does not necessarily need to be weighted. However, modelling uncertainty and input noise effects may be more serious in one or more frequency bands. The performance index should reflect this fact, and hence a frequency dependent weighting factor must also be placed on J2(K), in some situations.
J~(h') =
sup
-~{W2(jw)C(jwI - Ac) -1}
(104)
Now, considering the steady state value of the residual:
r__(oc) - f.,(cx~) + A-~lK f ~(oo) - A:l_d(oc)
(105)
After the transient period, the residual steady state value plays an important role in fault detection. Ideally, it should reconstruct the fault signal. The disturbance effects on residual can be made minimized by minimizing the following performance index:
J~(K) =11 A : 1 II
(106)
When J4 is minimized, the matrix Ii is very large and the norm II AY ~K II approaches a constant value. This means that the fault effect on the residual has not been changed by reducing the disturbance effect. This is what is required for good FDI performance. As there are four performance indices, to achieve robust fault detection (in terms of minimizing missed and false alarm rates), a multi-objective optimization problem must be solved. The parameter set to be designed
212
R.J. PATTONAND J. CHEN
are the elements of the gain matrix K, which must guarantee the stability of the observer. This leads to a constrained optimization problem which is difficult to solve. To tackle this problem, Chen et al [80,81] use the eigenstructure assignment method, via the dual control problem, to yield the parameterisation of the gain matrix K. The multi-objective problem has been solved using a combination of a genetic algorithm and the method of inequalities [21,80,81].
VIII. ROBUST
DECISION-MAKING
IN TO FDI
Efforts to enhance the robustness of FDI can be made at the decisionmaking stage [13,14,15,20,68,69]. Due to inevitable parameter uncertainty, disturbance and noise encountered in a practical application, one will rarely find a situation where the conditions for a perfectly robust residuM generation are met. This is especially true for unstructured uncertainties. It is therefore necessary to provide sufficient robustness not only in the residual generation stage but also in the decision-making stage. When the decisionmaking stage of FDI is made robust against uncertainty, we can speak of passive robustness in FDI [14,15], in which case it may not be necessary (or it may be difficult) to make the residual robust. Passive robustness is thus an alternative to active robustness which should be used when there is very limited plant information available. The goal of robust decision-making is to minimize the false and missing alarm rates due to the effects that modelling uncertainty and unknown disturbances will have on the residuals. This can be achieved in several ways, e.g. by statistical data processing, averaging, or by finding and using the most effective threshold. When a fixed threshold is used, the sensitivity to faults will be intolerably reduced if the threshold is chosen too high, whereas the false alarm rate will be too large when the threshold is chosen too low. The proper choice of the threshold is a delicate problem. Walker & Gai [82] and Walker [83] proposed the determination of the optimal threshold via Markov theory. In the case of large manoeuvres these changes might be large enough so that there is no fixed threshold that allows satisfactory FDI at a tolerable false alarm rate. A. A D A P T I V E
THRESHOLDS
The methods of passive robustness in FDI which have received the most attention are based on the use of adaptive thresholds [68,84-87], i.e. each threshold becomes a function in some way of measurable quantities. In the
ROBUST FDI SYSTEMS
213
adaptive threshold approach the residual thresholds are varied according to the control activity of the process. This concept is illustrated in Fig. 8 which also shows the typical shape of an adaptive threshold for direct residual evaluation. The idea of adapting the thresholds for fault detection, to the instantaneous operation of the monitored plant has been used by several investigators [68,84-87]. An interesting question is how should we determine the functional form of the adaptive threshold law?
Figure 8: Application of an adaptive threshold Clark [84] used an empirical adaptive law. Horak [86] calculated the envelope of maximum variations in an output based on an interval description of uncertain parameters (i.e. using bounds upon the uncertainty). He chose, at every time instant and for each parameter, the value within its uncertainty range that maximizes the output and also the parameter vector that would minimize the same output. Within this range of outputs, the variations can therefore be explained by the parameter uncertainty. In this way Horak used, at every time step, the smallest possible threshold required to prevent false alarms in the presence of the worst case possible parameter deviations. Whilst this idea is intuitively appealing, it is computationally very complex for all but simple examples of uncertain systems. Emami-Naeini et al [85] in their threshold selector (or threshold adaptor) method use the same strategy of characterising the set of detectable faults as those whose smallest possible effect on the norm of the residuals (under assumptions of bounds on norms of noise and plant uncertainty) are larger than the largest possible effect of noise and uncertainty alone. Ding & Frank [68] developed this concept further in connection with frequency domain approaches as follows. If the disturbance de-coupling condition in Eq.(40) holds true, the fault-free residual is:
(lO7)
214
R . J . PATTON A N D J. C H E N
Assuming that the modelling error is bounded by some limiting value 5, i.e.
ItAG~(s)]] <_ 5
(108)
In this situation, the fault-free residual will be bounded as"
II__r(s)ll- IlHu(s)zXG~,(s)u__(s)[I <_ allHu(s)u(s)l I
(109)
An adaptive threshold can be assigned as follows:
Th(s)
5Hy(s)u(s)
-
Isaksson [87] developed this a priori threshold selection into an scheme. His approach differs from the studies of Horak [86] and Frank [68] in the form of the assumed uncertainty description. of describing the uncertainty in terms of parameter intervals, he unstructured frequency domain bound:
(rmax[A(jw)] <_6(jw)
(110) on-line Ding ~z Instead uses an
(111)
where ~ma~[.] denotes the maximum singular value and the real (perturbed) plant transfer matrix is given by:
G(s)
-
(I + A(s))G~,(S)
(112)
and with Gu(s) the nominal plant transfer function. A similar approach has been described earlier by Weiss [88] who applied a bound on the uncertainty of the magnitude response, in order to construct time variable thresholds for fault detection. He assumed that an upper bound L(w) on the Fourier transform, E(w), of the difference in impulse response between the model and the real plant can be determined, usually considered as a high-pass filter, since the high frequency dynamics of the plant are often ignored in the model. Weiss used a test statistic in t:he form of a weighted sum of the open-loop residuals and compared this with the time-varying threshold as a weighted sum of the primary residual vector eu(t ) and the decision function f2(t)" _%(t)
-
y_(t) - ~(t)
(113)
t
Q(t)
-
E T'---- O 0
w(T)e_y(t- 7-)2
(114)
ROBUST FDI SYSTEMS
215
This is compared with the time-varying threshold"
Th (t) -- E
w(r -- t)
T----O0
l(q)u(t -- q)
(115)
q----Cx:)
where l(t) is the time domain counterpart of L(w) and w(t) provides the necessary forgetting. The method can be effective for slow changes, when a tight bound on the model is available in the low frequency range. At high frequencies it will anticipate larger model mismatch and therefore will tolerate larger errors. It is believed that a good solution to robust FDI is to combine the disturbance de-coupling design with adaptive threshold. B. R O B U S T
DECISION-MAKING
BASED
ON FUZZY LOGIC
When an FDI residual can be made robust against uncertainty (the socalled active approach), the process of deciding as to whether or not a fault has occurred in the system is facilitated. In this case, with fixed thresholds the false and missed-alarms rates are minimal. It is very difficult to achieve true robustness in residual generation and the active approach to robust FDI is an a t t e m p t to include robustness at the DECISION-MAKING stage. The use of adaptive thresholds is one way to achieve a low false alarm rate, as discussed in Section VIII.A.. The decision-making process, in one way or another, comes down to a comparison of a decision function with a threshold. When the threshold is a numerical value (either fixed or adaptive), it is said to be crisp. An alternative approach is to make use of fuzzy logic decision-making, as discussed by Frank and Kiupel [89]. Fault decisions based on fuzzy logic, provide a way of avoiding the effects of uncertainty and disturbance and hence reducing the false and missed alarm rate [89-91]. In the theory of fuzzy logic, fuzzy sets are, like crisp sets, defined on exact fundamental sets. For all values of the fundamental set, the membership function of a crisp set has only two elements, '0' and '1'. In contrast to the crisp set the membership function of a fuzzy set can take on all values between '0' and '1'. The value '0' stands for no membership, whilst the value '1' stands for a full membership. Thus the crisp borders of the classical sets vanish and are replaced by fuzzy sets.
IX. C O N C L U D I N G
DISCUSSION
This chapter has reviewed most of the latest advances in the field of robustness for fault diagnosis, based on model-based residual generation methods.
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R.J. PATTONAND J. CHEN
The study has drawn out the correspondences between various approaches connected with either parity space concepts or state observers (or both together). After outlining the basic properties of the main approaches, the general problem of robustness in FDI has been described as a foundation for discussing the ways in which robustness can be dealt with either in the frequency or time domains. Very few methods have actually made use of the time domain, however it has been shown that the eigenstructure is a very powerful tool for bridging the time and frequency domains to provide optimal disturbance de-coupling. The freedom available for residual generation design and for fault isolability is governed by rules governing the design freedom of any multivariable system. An observer can, of course, be considered as a dualization of the feedback control problem, for which there are several decades of multivariable design theory available. In order to understand how to aclfieve the best performance of an FDI scheme in terms of robustness to uncertainty, sensitivity to faults and fault isolability, it is essential to know something about the degrees of freedom available for these tasks, given a particular FDI approach. It is very clear from this study that the freedom issue has not been properly understood in the literature. It has also not been fully appreciated that the failure detection filter is no other than a full order observer with special directional s~..nsitivity properties for actuator or sensor fault isolation. When trying t(:, achieve good fault isolation with one observer (or failure detection filter), it is almost impossible to achieve good robustness to u n c e r t a i n t y - both calmot live together perfectly, when using a linear system approach! This takes us to the point that residuals can be designed using nonlinear theory, a non-linear observer, or non-linear feedback design, or just using non-linear or time-varying residuals. Clearly, if the system's dynamic structure and non-linearity are not well known then we are better off trying to use approximate uncertainty de-coupling methods. All real life systems have uncertain characteristics and cannot be modelled perfectly. The chapter has drawn out the essential differences between various methods for optimizing robustness, based on whether or not the uncertainty can be considered as structured or unstructured. In some cases, even systems with unstructured uncertainty can form the basis of estimation schemes generating an approximate structure, so that multivariable de-coupling theory can be used. Most of the robust FDI approaches based on unstructured uncertainty use frequency domain optimization and several papers have discussed the importance of various observer structures, within this framework. However, there has been a lack of understanding of the real point that fault detection and fault isolation is an input-output processing problem, which means that state variables and consequently, state
ROBUSTFDI SYSTEMS
217
estimates are really of very little consequence to the robust FDI problem. The chapter has pointed out, the importance of appropriate optimization and structure of residuals for robust fault detection and, to some extent, robust fault isolation (although these problems are not easily separable). It is clear that the robust FDI problem is not an issue of whether or not one should use an observer or a generalized parity space. Future work on robust FDI will continue to make use of optimization procedures, whether or not they are based upon H ~ . Of course, we will never understand the true value of robust FDI theory until the methods are applied to real process plants. Developments in real application studies are eagerly awaited.
IX. A C K N O W L E D G E M E N T S The authors wish to acknowledge funding support for this research from the UK Engineering & Physical Science Research Council through grants ref. GR/J12758 & GR/J12789.
References [1] A. S. Willsky, "A survey of design methods for failure detection in dynamic systems", Automatica 12 (6), pp.601-11 (1976). [2] R. Isermann, "Process fault detection based on modelling and estimation methods: A survey", Automatica 20, pp.387-404 (1984). [3] J. J. Gertler, "A survey of model-based failure detection and isolation in complex plants", IEEE Control Systems Magazine 8, (6), pp.3-11 (1988). [4] M. Basseville, "Detecting changes in signals and systems- A survey", Automatica 24 (3), pp.309-326 (1988). [5] R.J. Patton, P. M. Frank and R. N. Clark, Fault Diagnosis in Dynamic Systems: Theory and Application, Prentice Hall, (1989). [6] P. M. Frank, "Fault Diagnosis in dynamic system using analytical and knowledge based redundancy - A survey and some new results", Automatica 26 (3), pp.459-474 (1990). [7] R.J. Patton and J. Chen, "A review of parity space approaches to fault diagnosis", Proc. of IFAC/IMACS Symposium SAFEPROCESS'91, Baden-Baden, Sept.10-13, pp.239-255 (1991). [8] R. J. Patton, "Fault detection and diagnosis in aerospace systems using analytical redundancy", IEE Computing ~ Control Journal 2 (3), pp.127-136 (1991).
218
[9]
R.J. PATTONANDJ. CHEN
P. M. Frank, "Fault diagnosis in dynamic system via state estimation- A survey", in: System Fault Diagnostics, Reliability ~fi Related Knowledge-based Approaches, Tzafestas, Singh & Schmidt (eds), Vol. 1, pp.35-98, D. Reidel Press (1987). [10] E. Y. Chow and A. S. Willsky, "Analytical redundancy and the design of robust detection systems", IEEE Trans. Aut. Control AC-29 (7), pp.603-614 (1984). [11] J. J. Gertler, "Residual generation in model-based fault diagnosis", Control- Theory and Advanced Technology 9 (1), pp. 259-285 (1993). [12] G. G. Leininger, "Model degradation effects on sensor failure detection", Proc. of the 1981 Joint American Control Conf. Vol.2: Paper FP-3A, USA, June (1981) [13] P. M. Frank, "Enhancement of robustness in observer-based fault detection", Int. J. Control 59 (4), pp.955-981 (1991). [14] R. J. Patton and J. Chen, "Robustness in model-based fault diagnosis", In: Concise Encyclopedia of Simulation and Modelling (eds: D Atherton, P Borne), 379-392, Pergamon Press (1992). [15] R. J. Patton and J. Chen, "A survey of robustness in quantitative model-based fault diagnosis", Appl. Math. and Comp. Sci. 3 (3), pp.399-416 (1993). [16] R. J. Patton, "Robust model-based fault diagnosis: the state of the art", Proc. of IFA C/IMA CS Symposium SAFEPROCESS'94, Finland, June 13-16, Vol.1, pp.1-27 (1994). [17] K. Watanabe and D. M. Himmelblau, "Instrument fault detection in systems with uncertainties", Int.J. Syst. Sci. 13, pp.137-158 (1982). [IS] P. M. Frank and L. Keller, "Sensitivity discriminating observer design for instrument failure detection", IEEE Trans. A erosp. Electron. Syst. AES-16 (4), pp.460-467 (1980) [19] X. Lou, A. S. Willsky and G. C. Verghese, "Optimally robust redundancy relations for failure detection in uncertain systems", Automatica 22 (3), pp.333-344 (1986) [2o] P. M. Frank and X. Ding, "Frequency domain approach to optimally robust residual generation and evaluation for model-based fault diagnosis", Automatica 30 (4), pp. 789-804 (1994). [21] J. Chen, Robust Residual Generation for Model-based Fault Diagnosis of Dynamic Systems, DPhil Thesis, University of York, (1995). [22] P. M. Frank, "Advances in observer-based fault diagnosis", P~vc. of the Conf.: TOOLDIAG '93, Toulouse, France, April 5-8, pp.8].7-836 (1993).
ROBUSTFDI SYSTEMS
219
[23] R. Seliger and P. M. Frank, "Fault diagnosis by disturbance decoupled nonlinear observers", Proc. of the 30th IEEE Con. on Decision and Control, Dec 11-13, UK, pp.2248-2253 (1991). [24] D. L. Yu, D. N. Shields and J. L. Mahtani, "Fault detection for bilinear systems with application to a hydraulic system", Proc. of The 3rd IEEE Conf. on Control Applications, Glasgow, UK, August 24-26, pp.13791384 (1994). [25] J. Chen and R. J. Patton, "A re-examination of fault detectability and isolability in linear dynamic systems", Proc. of IFA C/IMA CS Symposium 2AFEPROCESS'94, Finland, June 13-16, pp.590-562 (1994). [26] R. N. Clark, "Instrument fault detection", IEEE Trans. Aero. 8J Electron. Syst. AES-14, pp.456-465 (1978). [27] J. Wiinnenberg, Observer-based Fault Detection in Dynamic Systems, PhD Thesis, University of Duisburg, Germany, (1990). [28] R. V. Beard, Failure Accommodation in Linear Systems Through SelfReorganization, PhD Thesis, MIT (1971). [29] J. E. White and J. L. Speyer, "Detection filter design: spectral theory and algorithm", IEEE Trans. on Auto. Control A C - 3 2 (7), pp.593-603 (1987). [30] Y. Park and G. Rizzoni, "An eigenstructure assignment algorithm for the design of fault detection filters", IEEE Trans. on Auto. Control 39 (7), pp.1521-1524 (1994). [31] R. K. Mehra and J. Peschon, "An innovations approach to fault detection and diagnosis in dynamic systems", Automatica 7, pp.637-640 (1971) [32] S. G. Tzafestas and K. Watanabe, "Modern approaches to system/sensor fault detection and diagnosis", Journal A 31 (4), pp.42-57 (1990). [33] L. A. Mironovski, "Functional diagnosis of dynamic system - A survey", Automn Remote Control 41, pp.1122-1143 (1980). [34] N. Viswanadham, J. H. Taylor and E. C. Luce, "A frequency-domain approach to failure detection and isolation with application to GE-21 turbine engine control systems", Control- Theory ~ Advanced Technology 3 (1), pp.45-72 (1987). [35] X. Ding and P. M. Frank, "Fault detection via factorization approach", Systems ~ Control Letters 14, pp.431-436 (1990). [36] R. Isermann, "Fault diagnosis of machine via parameter estimation and knowledge processing- tutorial paper", Automatica 29 (4), pp.815-835 (1993). [37] J. O'Reilly, Observers for Linear Systems, Academic Press, (1983).
220
R.J. PATTONANDJ. CHEN
[38] N. Viswanadham and R. Srichander, "Fault detection using unknown input observers", Control- Theory and Advanced Technology 3 (2) pp.91-101 (1987). [39] M. S. Phatak and N. Viswanadham, "Actuator fault detection and isolation in linear systems", Int. J. Syst. Sci., 19, pp.2593-2603 (1988). [40] R.J. Patton, "Robust fault detection using eigenstructure assignment", Proc. 12th IMACS World Congress Mathematical Modelling and Scientific Computation, Paris, July 18-22, Vol.2, pp.431-434 (1988) [41] R. J. Patton and S. M. Kangethe, "Robust fault diagnosis using eigenstructure assignment of observers", in Patton, Frank & Clark (Eds): Fault Diagnosis in Dynamic Systems: Theory and Applications, Chapter 4, pp.99-154, Prentice Hall (1989). [42] R. J. Patton and J. Chen, "Robust fault detection using eigenstructure assignment: A tutorial consideration and some new results", Proc. of the 30'th IEEE Conf. on Decision and Control, Brighton, UK, Dec. 11-13, pp.2242-2247 (1991). [43] E. G. Gai, M. B. Adams, B. K. Walker and T. Smestad, "Correction to determination of failure thresholds in hybrid navigation", IEs Trans. Aero. 8J Electronic Systems AES-14, pp.696-697 (1978). [44] K. C. Daly, E. Gai and J. V. Harrison, "Generalized likelihood test for FDI in redundancy sensor configurations", AIAA J. of Guidance 8j Control 2 (1), pp.9-17 (1979). [45] A. Ray and R. Luck, "An introduction to sensor signal validation in redundant measurement systems", IEEE Control Systems Magazine 11 (2), pp.44-49 (1991). [46] J.Gertler and Q. Luo, "Robust isolable models for failure diagnosis", AIChE J. 35 (11), pp.1856-1868 (1989). [47] J. Gertler, X. W. Fang and Q. Luo, "Detection and diagnosis of plant failures; the orthogonal parity equation approach", Leondes, C. (ed), Control ~ Dynamics Systems, Academic Press, Vol. 37, p1::,.157-216 (1990) [48] M. A. Massoumnia and W. E. Vander Velde, "Generating parity relations for detecting and identifying control system component failures", J. Guidance, Control 8J Dynamics 11 (1), pp.60-65 (1988). [49] J. Gertler and M. M. Kunwer, "Optimal residual decoul)ling for robust fault diagnosis", Proc. of Int. Conf. on Fault Diagnosis: TOOLDIAG'93, Toulouse, France, April 5-7, pp.436-452 (1993). [50] M. A. Massoumnia, A geometric approach to failure detection and identification in linear systems, PhD Thesis, Dept. of Aero. & As~ro. MIT (1986).
ROBUSTFDI SYSTEMS
221
[51] R. J. Patton and J. Chen, "A re-examination of the relationship between parity space and observer-based approaches in fault diagnosis", European Journal of Diagnosis and Safety in Automation (Revue europdenne Diagnostic et s~tretd de fonctionnement) 1 (2), pp.183-200
(1991).
[52] P. M. Frank and J. Wiinnenberg, "Robust fault diagnosis using unknown input schemes", in Patton, Frank ~ Clark (Eds): Fault Diagnosis in Dynamic Systems; Theory and Applications, Chapter 3, pp.47-98, Prentice Hall (1989). [53] R. J. Patton and J. Chen, "A review of parity space approaches to fault diagnosis for aerospace systems", AIAA J. Guidance, Control Dynamics 17 (2), pp.278-285 (1994). [54] H. J. Marquez and C. P. Diduch, "Sensitivity of failure detection using generalized observers", Automatica 28, pp.837-840 (1992). [55] Y. X. Yao, J. Schaefers and M. Darouach, "Robust fault detection based on factorization approach", Proc. of IFA C/IMA CS Symposium SAFEPROCESS'94, Finland, June 13-16, Vol.2, pp.525-530 (1994). [56] M. Vidyasagar, Control Systems Synthesis: A Factorization Approach, MIT Press, North Holland System and Control Series (Vol.8)", Cambridge, MA, (1985). [57] R. Da, "Failure detection of dynamical systems with the state X2 test", AIAA Y. Guidance, Control ~ Dynamics 17 (2), pp.271-277 (1994). [58] S. Tanaka and P. C. Mfiller, "Fault detection in linear discrete dynamic systems by a pattern-recongnition of a Generalized-Likelihood-Ratio", J. Dyn. Syst., Meas. 8J Control- Trans. of the ASME 112 (2), pp.276-
282 (1990).
[59] K. A. Loparo, M. R. Buchner and K. S. Vasudeva, "Leak detection in an experimental heat exchanger process: A multiple model approach", IEEE Trans. Aut. Control AC-36 (2), pp.167-177 (1991). [60] B. Friedland and S. Grabousky, "Estimating sudden changes of biases in linear dynamic systems", IEEE Trans. Aut. Control AC-27 (1), pp.237-240 (1982). [61] J. Chen, H. G. Zhang and H. Y. Zhang, "A modified separated-bias estimation approach to the detection and estimation of failures in linear systems", Preprints of 1990 XI IFA C World Congress, Tallin, August
(1990).
[62] W. Ge and C. Z. Fang, "Detection observation", Int. Y. Control 47 (2), [63] W. Ge and C. Z. Fang, "Extended failure isolation", Int. J. Control 49
of faulty components via robust pp.581-599 (1988). robust observation approach for (5), pp.1537-1553 (1989).
222
[64] [65] [66]
[67]
[68]
[69] [70] [71]
[72]
[73]
[74]
[75] [76]
R.J. PATTON AND J. CHEN
J. Chen and H. Y. Zhang, "Robust detection of faulty actuators via unknown input observers", Int. J. of Sys. Sci. 22 (10), pp.1829-1839 (1991). R. J. Patton and J. Chen, "Robust fault detection of jet engine sensor systems by using eigenstructure assignment", J. Guidance, Control Dynamics 15 (6), pp.1491-1496 (1992). R. J. Patton and J. Chen, "Optimal Selection of unknown input distribution matrix in the design of robust observers for fault diagnosis", Automatica 29 (4), pp.837-841 (1993). N. Viswanadham and K. D. Minto, "Robust observer design with application to fault detection", Proc. the 1988 Amer. Control Conf., Atlanta, GA, Vol.3, pp.1393-1399 (1988). X. Ding and P. M. Frank, "Frequency domain approach and threshold selector for robust model-based fault detection and isolation", Proc. of IFAC Symposium SAFEPROCESS'91, Baden-Baden, Sept. 1991, pp.307-312 (1991). P. M. Frank and X. Ding, "Frequency domain approach to minimizing detectable faults in FDI systems", Appl. Math. 8J Comp. Sci. 3 (3), pp.417-443 (1993). Z. Qiu and J. J. Gertler, "Robust FDI systems and H~-optimization: Disturbances and tall fault case", Proc. of The 32nd IEEE Conf. on Decision 8J Control, Texas, USA, Dec. 15-17, pp.1710-1715 (1993). X. Ding and P. M. Frank, "Fault detection via optimally robust detection filters", Proc. of the 28lh IEEE Con. on Decision and Control, Tampa, Dec. 13-15, pp. 1767-1772 (1989). R. J. Patton, J. Chen and H. Y. Zhang, "Modelling methods for improving robustness in fault diagnosis of a jet engine system", Proc. of the 31th IEEE Conf. on Decision CJ Control, Tucson, Arizona, Dec.1618, pp.2330-2335 (1992). J. Gertler, "Modelling errors as unknown inputs", Proc. of IFA C/IMA CS Symposium SAFEPROCESS'94, Finland, June 13-16, Vol.1, pp.266-271 (1994). M. Hou and P. C. Miiller, "Disturbance decoupled observer design: A unified viewpoint", IEEE Trans. on Auto. ControlAC-39 (6), pp.13381341 (1994). M. Hou and P. C. Miiller, "Fault-detection and isolation observers", Int. J. Control 60 (5), pp.827-846 (1994). J. Chen, R. J. Patton and H. Y. Zhang, "Design of unknown input observers and robust fault detection filters", accepted for publication by Int. J. Control, (1995).
ROBUST FDI SYSTEMS
223
[77] N. Karcanias and B. Kouvaritakis, "The output-zeroing problem and its relationship to the invariant zero structure", Int. J. Control 30, pp.395-415 (1979). [78] P. M. Olin and G. Rizzoni, "Design of robust detection filters", Proc. of 1991 American Control Conf., pp.1522-1527 (1991). [79] M. Saif and Y. Guan, "A new approach to robust fault-detection and identification", IEEE Trans. Aerosp. Electron. Syst. AES-29 (3), pp.685-695 (1993). [80] J. Chen, R. J. Patton and G. P. Liu, "Design of optimal residuals for detecting sensor faults using multi-object optimization and genetic algorithms", Proc. of 1994 AIAA Guidance, Navigation, and Control Conf., August 1-3, The Phoenician Scottsdale, AZ, pp.349-357 AIAA94-3581-CP (1994). [81] J. Chen, R. J. Patton and G. P. Liu, "Detecting incipient sensor faults in flight control systems", Proc. of The Third IEEE Conference on Control Applications, August 24-26, Glasgow, Scotland, Vol.2, pp.871876 (1994). [82] B. K. Walker and E. Gai, "Fault detection threshold determination techniques using Markov theory", J. Guidance, Control ~ Dynamics 2 (4), pp.313-319 (1979). [83] Walker B (1989), "Fault detection threshold determination using Markov theory", in Patton, Frank & Clark (Eds): Fault Diagnosis in Dynamic Systems: Theory and Applications, Chapter 14, pp.477-508, Prentice Hall (1989). [84] R. N. Clark, "State estimation schemes for instrument fault detection", in Patton, Frank & Clark (Eds): Fault Diagnosis in Dynamic Systems: Theory and Applications, Chapter 2, pp.21-45, Prentice Hall (1989). [85] A. E. Emami-Naeini, M. M. Akhter and S. M. Rock, "Effect of model uncertainty on failure detection: the threshold selector", IEEE Trans. Aut. Control AC-33 (2), pp.1106-1115 (1988). [86] D. T. Horak, "Failure Detection in Dynamic Systems with Modelling Errors", J. Guidance, Control and Dynamics 11 (6), 508-516 (1988). [87] A. J. Isaksson, "An on-line threshold selector for failure detection", Proc. the Int. Con.: TOOLDIAG'93, Toulouse, CERT-ONERA, pp.628-634 (1993). [88] J. L. Weiss, "Threshold computations for detection of failures in SISO systems with transfer function errors", Proc. of the 1988 American Control Conf., pp.2213-2218 (1988). [89] P. M. Frank and N. Kiupel, "Fuzzy supervision and application to lean production", Int. Systems Sci. 24 (10), pp.1935-1944 (1993).
224
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[90] P. M. Frank, "Application of fuzzy logic process supervision and fault diagnosis", Proc. of IFA C/IMA CS Symposium SAFEPROCESS'94, Finland, June 13-16, Vol.2, pp.531-538 (1994). [91] M. Ulieru and R. Isermann, "Design of fuzzy-logic based diagnostic model for technical process", Fuzzy Sets and Systems 58 (3), pp.249271 (1993).
ABSOLUTE STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS
YASUHIKO
MUTOH
TIELONG
SHEN
D e p a r t m e n t of Mechanical Engineering Sophia University Chiyoda-ku, Tokyo, J a p a n
PETER
N. N I K I F O R U K
D e p a r t m e n t of Mechanical Engineering University of Saskatchewan Saskatoon, Saskatchewan, C a n a d a
I. I N T R O D U C T I O N The absolute stability of discrete-time nonlinear feedback control systems is discussed in this chapter. In this regard, during the recent past several different types of stability theory of nonlinear control system have been put forth [2] [3] [4]. In particular, the differential geometric approach has been well developed, which allows the design of stable nonlinear control systems using exact linearization, or input-output linearization technique if the nonlinear plant satisfies a certain condition. Although this approach is theoretically very attractive, control problems with sector nonlinearities in the feedback loop such as hard spring, soft spring, dead zone, saturation, etc. still have to be dealt with. Hence, from a practical point of view, since the stability of control systems with these types of nonlinearities is still CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
225
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YASUHIKO MUTOH ET AL.
very important, attention is focused in this chapter here on the stability problem of this type of such nonlinear control systems. Control problems with sector nonlinearities are kind of Lurie problem and have been considered for some time using the concept of the absolute stability. The most powerful and well known stability condition for such nonlinear control systems is the circle criterion. This gives a sufficient condition for a feedback control system to be asymptotically stable in the large for such a nonlinearity that belongs to some prespecified class. On the other hand, the absolute stability problem can be discussed from several different points of view. In this chapter, three approaches to this problem will be considered. The first one is the ordinary circle criterion which will be discussed using the Lyapunov function. The second is the contraction mapping technique which will be used for the absolute stability problem. In this discussion, the determination of a sector that gllarantees the global asymptotic stability is regarded as the problem of the simultaneous Lyapunov problem [7]. Using this fact, the relation between the stability region of a sector for the nonlinear and a linear parts of the feedback system will be derived. Finally, the absolute stability of multi-input multi-output nonlinear feedback systems is considered. In this discussion, the H ~ norm of a transfer matrix and the matrix Riccati inequality will be used to analyze the absolute stability, which is the generalization of the circle criterion for multivariable systems. Since the absolute stability problem can be regarded as a robust stability problem with structured uncertainty, the relation between the quadratic stability and the circle criterion of multivariable systems is also considered. Finally, the exponential convergence property will be discussed. This final discussion will allow an explanation of the absolute stability problem from very general point of view.
II. STATEMENT OF THE P R O B L E M Consider a single-input single-output discrete-time nonlinear feedback system E, shown in Fig. 1, which consists of linear dynamic elements in the forward path and a memoryless nonlinearity ~(y(t), t) in the feedback path described by the following equations. Linear P a r t :
x(t + 1) = y(t)
=
Ax(t) 4- bu(t) eTa(t)
(1)
STABILITYOF DISCRETENONLINEARFEEDBACKSYSTEMS
227
Nonlinear Part :
u(t) = - p ( y ( t ) , t )
(2)
where z(t), b, c E R n, y(t), u(t) E R 1, A E R n• and t is an integer. R n and R n • denote the n-dimensional real vector space and n • n real matrix space, respectively. In the linear dynamic part, the matrix A is assumed to be stable and eq. (1) is assumed to be controllable and observable. Further, it is assumed that ~ ( 0 , t ) = 0 and ~(y(t), t ) l i e s in a sector
kL <
t) < ku.
(3)
y(t) This implies that ~(y(t), t) lies in the sector between ~ = k L y and ~ = k u y , as shown in Fig. 2, which is typical of practical nonlinearities (Fig. 3). For simplicity, this sector is denoted by [kn, ku] and eq. (3) can be written as
(4)
(~(y(t), t) - k u y ( t ) ) ( ~ ( y ( t ) , t) - kLy(t)) < O.
Since 9(0, t) = 0, x = 0 is an equilibrium point of the nonlinear feedback control system, and the problem, then, is to investigate the stability of this equilibrium point. For this purpose, define then the absolute stability of the nonlinear feedback system and the Hurwitz sector by the following.
-t-
)
~._ x(t+ 1) = Ax(t)+bu(t) y(t) =crx(t)
0
y
Fig.1 Nonlinear Feedback System
Y
cs~
~,.
o j.., ~-,~
~'.
J~
C~
~ (TQ
o
~ v
~
~ C~. v
~b,b
~
'b'b~%b j
~hl ~
~. 0
~ c~
oN
I~ v
zo
o~
c~ ~.
II
II
STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS
229
Definition 1 The nonlinear feedback system E is said to be absolutely stable with respect to the sector [kL, ku] if x = 0 is asymptotically stable in the large for all nonlinearities satisfying eq. (3). | Definition 2 A sector [kn, ku] (kn < kcr) is called the Hurwitz sector of the system eq. (1) if A - beTk is asymptotically stable for all k E [kn, kv]. g
Before the circle criterion was established, there was a stability hypothesis known as Aizerman's conjecture for continuous time systems [5]. This conjecture was that if a nonlinearity ~(y(t), t) lies in a Hurwitz sector, the nonlinear feedback system $] may be absolutely stable. Unfortunately, the situation is not so simple [6]. From this point of view, it is interpreted that the absolute stability problem is to find such a sector [kn, ku] where Aizerman's conjecture is valid if the Hurwitz sector is replaced by the sector [kL, ku]. In the sequel, three approaches to this problem will be considered for discrete-time control systems.
III. CIRCLE CRITERION In this section, a summary is given of the circle criterion for single-input single-output discrete-time systems, which is the most orthodox approach for determining absolute stability. Since the concept of the positive realness of a certain transfer function plays an important role, some facts about the positive realness should be noted first. Definition 3 The real rational function G(z) is called positive real if (1) G(z) is analytic in ]z[ > 1 (2) ReG(z) > 0 in Izl > 1.
Condition (2) of Definition 3 can be replaced by
(2)' ReC(
> 0w
[0, 2 1.
Definition 4 G(z) is called strictly positive real if G(pz) is positive real for some positive number p < 1. | The following Lemma shows the connection between the positive reap ness in the frequency domain and in the time domain.
230
YASUHIKOMUTOHET AL.
r =ky ) 9
0
Fig.4
r
y
Sector Nonlinearity
L e m m a 1 Let {A, b, cT, d} be a minimum realization of H(z), i.e., H(z) - cT(zt-
A ) - l b + d.
(5)
Then, H ( z ) is strictly positive real if and only if there exist real matrices p _ p T > 0 and L and constants w and 7 such that A TPA-
P
=
- L L T - 72I
A TPb
-
c+wL
w2
_
2 d - bT Pb.
(6)
|
Before considering general sector nonlinearities, the stability of a nonlinear feedback system with a particular nonlinearity ~ ( y ( t ) , t ) belonging to the sector [0, k] (k > 0) in Fig. 4, will be discussed, i.e., t) - k u ( t ) ) <_ o.
We then have the following result.
(7)
STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS
231
T h e o r e m 1 The nonlinear feedback system E is absolutely stable with respect to the sector [0, k] if G(z) + ~ is strictly positive real, where G(z) cT(zIA)-ib. | (Proof) From Lemma 1, if G ( z ) + ~ is strictly positive real, there exist real matrices P - p T > 0 (E R nxn), a vector h (E R n x i ) and constants w and 7 such that
A TPA-
P
ATpb w2
=
_ h h T _ 721
-
c+wh
-_
2 _ bT pb. k
(8)
Using this positive definite matrix P, let V(x(t)) be a candidate of a Lyapunov function defined by
(9)
V(x(t)) - x T (t)Px(t). Then,
:v(~(t))
= =
V ( x ( t + 1 ) ) - v(x(t)) (Ax(t) - b~(y(t), t)) T P ( A x ( t ) - b~(y(t), t)) - x T ( t ) P x ( t ) x T(t)(A T P A - P)x(t) - 2x T ( t ) A T Pb~(y(t), t) +~2(y(t),t)bTpb
=
~(t)(-hh
~ - ~I)x(t)
- 2x ~ ( t ) ( ~ +
,~h)~(y(t), t)
+~2(Y(t), t)(-~2 - w 2) --
--72X T ( t ) x ( t ) -- (h T x ( t ) -[- wy(y(t); t ) ) 2
2
+~(u(t),
t ) ( ~ ( u ( t ) , t) - k u ( t ) ) .
(10)
AV(z(t)) < O.
(11)
From eq. (7), This implies that V(x(t)) is monotonically decreasing as t ~ ~ , and, thus, AV(z(t)) ~ 0 as t ~ co. From eq. (10), this means that x(t) ~ 0 as t ~ :xD, which proves the Theorem. (Q.E.D.) The sufficient condition for the absolute stability of a nonlinear feedback system E with a more general sector nonlinearity can be stated as a Corollary of this Theorem.
232
YASUHIKOMUTOHET AL.
= ( ~ - kL)y
qg(y)- kt.y
S
0
y
Fig.5 Equivalent Sector for a General Sector Nonlinearity
C o r o l l a r y 1 The nonlinear feedback system ~2 is absolutely stable with respect to the sector [kL, ku] (kL < ku) if
1 + kuG(z) W(z) = 1 + kLG(Z)
(12)
1 + kuG(z) 1 G(z) kv - kL 1 + kL G(z) = k~ - k-----~+ a + kLC(Z)
(la)
kv - kL > 0
(14)
is strictly positive real. (Proof) From the relation 1
and the positive realness of W(z) and the positive realness of the second term of the right hand side of eq. (13) are equivalent. Note that the realization
C(z)
of 1 + kLG(z) is {A - bcTkL,b,c}. The nonlinear feedback system E with
STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS
233
a nonlinearity lying in the sector [kL, ktr] is represented as follows " x(t + 1)
y(t)
= Ax(t)-b~p(y(t),t) = ( A - bcTkL)x(t)- b(~(y(t),t)- kLcTx(t)) = ( A - bcTkL)x(t)- b(ta(y(t),t)- kLy(t)) = cTx(t).
(15)
Since the nonlinearity ~p(y(t), t) lies in the sector [kL,ktr], the nonlinearity ~p(y(t),t)- kLy(t) lies in the sector [0, kv - kL]. Thus, the nonlinear feedback system with a general sector nonlinearity is regarded as the nonlinear feedback system described in Fig. 5. Then, Corollary 1 is obtained from Theorem 1. (Q.E.D) Corollary 1 is the well known circle criterion for the absolute stability with respect to the sector [kL, kv]. The absolute stability of the nonlinear feedback system with respect to a given sector [kL,kv] can be verified by checking the stability of W(z) and examining the Nyquist plot of W(eJ~). However, since the circle criterion is the sufficient condition, even if Re[W(eJ~)] enters the left half plane, it does not necessarily means that the system E is unstable. The reason why Corollary 1 called the circle criterion is that, if G(z) is stable, Corollary 1 can be stated as the following Theorem. Let C ( - F u1 , 1 -- 2--~u 1 ) + 0j and whose radius is -~!-)_~ be a disk whose center is at ( - 2-~L
1 1
1
T h e o r e m 2 The nonlinear feedback system E with stable G(z) is absolutely stable if one of the following condition is satisfied according to the signs of kL and ku. 1 1. If 0 < kL < k~, the Nyquist plot of G(e j~) neither enters C(-~-~-u, - ~ )1 nor encircles C ( - ~--~,--~--[L 1 1 ) (Fig. 6 (a)). 2. If 0 = kL < ktr, the Nyquist plot of G(e j~) lies to the right of the line Z
--
--
1 (Fig 6 (b))
~U
9
.
3. If kL < 0 < ku, the Nyquist plot of G(e j~) lies in the interior of the
1 - k-i) 1 (Fig. 6 (c)) . disk C ( - k--~u, 4. If kL < k~ = 0, the Nyquist plot of G(e j~) lies to the left of the line 1 (Fig. 6 (d)).
234
YASUHIKOMUTOHET AL. 5. If kL < ku < 0, the Nyquist plot of G(e j'') neither enters C ( - ~ -1, 1 1 --FLL) nor encircles C(-~---~u , - ~ )1
Example
(Fig. 6 (e)).
1
Consider the following plant -
z2 + 0.3z - 0.4
and a sector [kL,kU] with kL -- --0.29 and ku - 1.2. This ~s case 3 1) of Theorem 2. The Nyquist plot of G(e j'') and the disk C ( - E T1u , - ~ - L ,
4
= C(-T~, ~) are depicted in Fig. 7, from which it is seen that the nonlinear feedback system of this example is absolutely stable. Fig. 7 shows that there still is a little more room left for extending this sector. 1 Z ~
~'~U
//".
~,-K) c(- 1 1 l
l
(a)
(b)
(c)
1
c(-~
1 ,
Q (o)
(e) Fig.6 Circle Criterion
, t - 1kL'
~
STABILITY OF DISCRETE NONLINEAR FEEDBACKSYSTEMS
Z
|
.w.~
i
i
!
235
'
.................................. I:S ii!i
< e~ i.-4
-1 -2 -3 -2
-1
0
1
2
3
4
5
Real Axis
Fig.7
Circle Criterion (Example 1)
IV. CONTRACTION MAPPING APPROACH A. STABILITY CONDITION In this section, the concept of contraction mapping will be used to analyze the absolute stability of nonlinear feedback systems. In the sequel, llxll is the Euclidean norm of a vector x and llAll is a norm of a m a t r i x A defined by the m a x i m u m singular value of A, (,~ma=(ATA)) 89 D e f i n i t i o n 5 [1] A function f(x) is called a contraction mapping in a region [2 (containing the origin) of the state space if Ilf(z)l I < Ilxll for all z in [2 with the exception of the origin.
(16) |
236
YASUHIKOMUTOHET AL.
L e m m a 2 [1] The null solution of the autonomous nonlinear system x(t + 1 ) = f ( x ( t ) )
(17)
with f(0) = 0, is asymptotically stable if f ( x ) is a contraction mapping in some neighbourhood of the origin of the state space. Further, it is asymptotically stable in the large if the function f ( x ) is a contraction in the whole state space. | D e f i n i t i o n 6 Consider a linear system {A, b, cT). A set of real numbers S ( P ) is called a set of simultaneous Lyapunov feedback gains if there exists a positive definite matrix P ( = p T ) such that (A - bcTk) T P ( A - b,;Tk) - P is negative definite for all k E S ( P ) . I This means that x T ( t ) P x ( t ) can be taken as a Lyapunov function for a system described by x(t + 1) = ( A - bcTk)x(t) for all k E S(P). Using this Definition, some particular type of a sector is defined as follows. D e f i n i t i o n 7 If a closed interval [kL, ku] is a set of a simultaneous Lyapunov feedback gains, a sector [kL, ku] is called a simultaneous Lyapunov sector. I The following Theorem shows that Aizerman's conjecture is valid in discrete time systems if the Hurwitz sector is replaced by a simultaneous Lyapunov sector. T h e o r e m 3 Consider the nonlinear feedback system E. If the nonlinearity ~(y) lies in a simultaneous Lyapunov s e c t o r []eL, kU], then the origin of such a nonlinear system is always asymptotically stable in the large. | (Proof) For all k E [kL, ku], there exists a symmetric positive definite matrix P such that (A - bcT k) T P ( A - be T k ) - P < 0 (18) It follows then that (A* - b" c *T k) T (A* - b* c "T k) - I < 0
(19)
where A*
-
T A T -1
b* c*T
_
Tb cTT
(20)
and T is a nonsingular matrix defined by P - TTT
(T
9 nonsingular)
(21)
STABILITYOF DISCRETENONLINEARFEEDBACKSYSTEMS
237
This implies that {A, b,c T} can always be transformed to {A*, b*,c *T} so that []A*-b*c*Tkll is less than 1. Hence, if k E [kL, ku], it can be assumed that ][A - bcT k[I < 1 (22) without loss of generality. On the other hand, the nonlinear system E can be written as
x(t + 1) F(x(t))
=
=
F(x(t)) Ax(t) - b~(cTx(t)).
(23)
Since p(c Tx(t)) lies in the sector [kL, ku], there exists some constant k~ E [kL, ku] such that ~(c T x(t)) = k~c T x(t) (24) which means that
F(x(t)) = (A - bcT k~)x(t).
(25)
[]g(x(t))[] < ]]x(t)[].
(26)
From eq. (22), Thus, F(x(t)) is a contraction mapping in the whole state space. Then, from Lemma 2, the nonlinear feedback system E is absolutely stable with respect to the simultaneous Lyapunov sector [kL, ku]. (Q.E.D.) This method is equivalent to the Lyapunov's method with xT(t)Px(t) as the Lyapunov function for the nonlinear feedback system E. The direct relation between the positive definite matrix P and the simultaneous Lyapunov sector will be discussed in the following section.
Bo EVALUATION OF THE SIMULTANEOUS LYAPUNOV
SECTOR In this section, a method is presented for evaluating the range of the simultaneous Lyapunov feedback gains. The plant is described by {A, b, cT } and A is assumed to be asymptotically stable without loss of generality. If this is not the case, A can be stabilized by a suitable state feedback. Then, there exists a symmetric positive definite matrix P which satisfies
A TPA - P = -Q
(27)
where Q is a given arbitrary symmetric positive definite matrix. The simultaneous Lyapunov problem is to find kL and kt; such that
( A - b c T k ) T p ( A - bcTk) - P < O,
vk
(2s)
238
Y A S U H I K O M U T O H ET AL.
Let {A, b, CT} be described in the observability canonical form, i.e.,
a]
0 A
E
cT
-
aER n
[0,...,0,1].
(29)
P can be partitioned as follows"
I* 1 $
$
...
:#
(30)
pn_~ p~,~
where P~ e
~r215
,
P~ i, P~ i
P~ E
R (n-i)x(n i) Pn-1 E R ~ - i , and ,
Pn,n E R i. Using these submatrices of P and eq. (29), the left hand side
of eq. (28) becomes -((A-
bcTk)Tp(A-
_ [ Pn-ll"
bcTk)-
P) pn_l-P2(a-bk) ] (31) Pn,n - (a - bk) i ' P ( a - bk)
Pd _ (a _ b k ) T p f
where Pd (E R (n-i)x(n-i))is defined by
p~ - p~2~_ p ~ .
(32)
The problem is to find the set of k such that eq. (31) is positive definite. Since, from eq. (27), the right hand side of eq. (31) is positive definite when k - 0, Pd is also positive definite. Hence, the desired range. [kn, ku] is such that det
Pd T _ (a bk)TP T Pn-1
-- (detPd) . [Pn,n - ( a -
Pn-1 -- P2(a - bk) ] Pn,n - (a - bk) T P ( a - bk) ] bk) T P ( a - bk)
- { P n - 1 - P2(a - b k ) } T P d l { P n - 1
- P2(a - blc)}]
= ( d e t P d ) . {d2k 2 + dl k + do}
>o
Vk ~ [kL, ku]
(33)
STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS
239
where
d2
-
- b T(P T PT PdlP2)b
dl
-
2{aT(p + P f P d l P 2 ) b - bTpTPdlpn-1} Pn,n - aT (p + pT Pdl P2)a - p nT_ l P d l p n _ l
do -
-t-2aT pT Pdlpn_ 1.
(34)
Since detPd > 0, kL and ku can be obtained as two real roots of the quadratic equation q(k) =
+
+ do = 0.
(35)
Since A is assumed to be stable, q(0) > 0, which implies that do > 0. From this and the fact that d2 < 0, eq. (35) always has two real roots. This means that there always exists a set of simultaneous Lyapunov feedback gains and it is a connected closed interval [kn, ku] determined by the two real roots of q(k), kn and ku. The simultaneous Lyapunov sector obtained here is the maximum sector for a certain choice of the Lyapunov matrix equation, or more specifically for an arbitrary positive definite matrix Q in eq. (27), which means that the thus obtained sector is not necessarily the maximum simultaneous Lyapunov sector for the given nonlinear feedback system. However, it should be emphasized that the simultaneous Lyapunov sector [kL, ku] and corresponding positive definite matrix P for the Lyapunov function xT(t)Px(t) for a given nonlinear feedback system can be obtained at the same time. The following example illustrates this. Example 2 Consider the same plant of Example 1 described by
G(z)-
1
2
9
z + 0.3z - 0.4
(36)
A corresponding realization in the observability canonical form is
A_[0 04] 1
-0.3
1
cT
=
[0,1].
(37)
Note that the Hurwitz sector for this case is [-0.3, 1.4]. In the following, three sets of the simultaneously stabilizing feedback gains are evaluated according to the choice of a symmetric positive definite matrix Q.
240
YASUHIKO MUTOH ET AL.
1) Q-
[10 o] ~_[ -0.921 284, 1
-0.921 ] 1.841
(38)
a simultaneously stabilizing gain sector :
2)
[kL, kv] = [-0.263, 1.029]
(39)
] Q_[21 31] ' ~_[ -0.651 6.635 -0.651 4.635
(40)
a simultaneously stabilizing gain sector : (41)
[kL, kU] = [-0.297, 1.225]
l tp
qg(y)= 1.4y Hurwitz Sector Simultaneous Lyapunov Gain Sector qgfy)=1.225y ~y) = -0.297y =
Fig.8 Hurwitz Sector and Simultaneous Lyapunov Sector for Example 1
-0.3y
STABILITY OF D I S C R E T E N O N L I N E A R F E E D B A C K S Y S T E M S
241
3) Q-
[5 2] 2 1 '
[6.587 2.540
P-
2.540] 1.587
(42)
a simultaneously stabilizing gain sector :
[kL, ku] = [-0.084, 0.301]
(43)
Among the above cases, since the second sector is the largest, it should be taken as a set of the simultaneously stabilizing feedback gains. Fig. 8 shows the Hurwitz sector and the simultaneously stabilizing gain sector for this plant. The corresponding positive definite matrix P is given by eq. (40) for the Lyapunov function of this nonlinear feedback system. Fig. 9 shows the contour of this Lyapunov function in the state space. It must be noted that another simultaneous Lyapunov sector may exist which contains those obtained here.
/
:
\i
eq
i i
!
O
!
//, X1
Fig.9 Contour of xT Px (Case 2)
242
YASUHIKO MUTOH ET AL.
V. ALGEBRAIC RICCATI INEQUALITY APPROACH
In this section, the absolute stability of multivariable systenls is discussed based on algebraic Riccati inequality approach. It will be shown that for the absolute stability of multivariable systems, a certain transfer matrix must satisfy some H ~ -norm condition, which can be regarded as the general circle criterion for the multivariable case. The relation between the absolute stability and quadratic stability will be shown also as will the fact that if the absolute stability is guaranteed by the Hor norm condition, the stability implies the exponential stability.
A. ABSOLUTE STABILITY OF MULTIVARIABLE SYSTEM Consider a multi-input and multi-output discrete time nonlinear feedback system ~ M described by the following equations. Linear Part: z(t+l) y(t)
= =
Ax(t)+Bu(t) Cx(t) + Du(t)
(44)
Nonlinear Part: u(t) = - ~ ( y ( t ) , t )
(45)
where u(t), y(t) E R m, A E R nxn, B E R '~xm,C E R mxn and D E R mxm. It is assumed that A is stable and the linear part is controllable and observable. The nonlinear function ~ : R m x R --~ R m is assumed to satisfy p(O, t) = O, and p(g(t), t) is said to lay in sector [kL, ku], if p(y(t), t) satisfies the following quadratic constraint {~(y(t), t) - kLg(t)} T {p(y(t), t) - f u y ( t ) } <_ O.
(46)
This constraint condition is a natural extension of the sector condition eq. (4) for the multivariable system. As the case of single-input singleoutput systems, Definition 1 can be used to define the absolute stability of multivariable systems by replacing E and eq. (4) by EM and eq. (46) respectively. Let G(z) = C ( z I - A ) - I B + D. The H,o norm of G(z) is defined by IIG(z)[I~ "- sup~e[0,2~ ] ~r(G(eJ~)), where a(.) denotes the maximum singular value. The following fact will be useful in the latter, which is the discrete time version of the result in H ~ control theory (See [9], [8]).
STABILITYOF DISCRETENONLINEARFEEDBACKSYSTEMS
243
L e m m a 3 A is stable and IIa(z)ll~ < ~ if and only if there exists a positive definite matrix P such that N "- 7 2 I - D T D - B T P B > 0 and the Riccati inequality - p+ATpA+CTC+(ATpB+CTD)N-I(BTpA+DTC)
< O
(47)
holds.
!
The following result gives a sufficient condition for the absolute stability of the system NM with respect to a particular sector I-k, k](k > 0). T h e o r e m 4 System NM is absolutely stable with respect to the sector 1
[-k, k] (k > O)if [[a(z)llo, < ~.
!
(Proof) Note that since kv - --kL -- k the quadratic constraint eq. (46) becomes ~T(y(t), t)~(y(t), t) <_ k~yT(t)y(t). (48)
1
Suppose that IIG(z)ll~ < ~. Then from Lemma 3, there exists a positive 1 definite solution P > 0 satisfying the Riccati inequality eq. (47) with 7 - ~. Consider the quadratic function V ( x ( t ) ) " - xT(t)Px(t) as a candidate of Lyapunov function for system EM. Then,
AV(x(t))
= :
V(x(t + 1 ) ) - V(x(t)) {Ax(t) - Bp(y(t), t)} TP{Ax(t) - BT(y(t), t)} --xT(x)Px(t) X T (t)(A T PA - P)x(t) - 2~ T (y(t), t)B T PAx(t) +pT (y(t), t)B T PB~(y(t), t). (49)
By substituting eq. (47) into eq. (49) and completing the squares gives
AV(x(t))
<
- x T ( t ) ( A T p B + C T D ) N - I ( B T p A + DTC)x(t) - - X T (t)C T Cx(t) - 2r T (y(t), t)B T PAx(t) +~T (y(t), t)B T PB~(y(t), t)
=
- I [ N - ~ (BTPA + D T C)x(t) + N 89p(y(t), t)ll 2 - x T (t)C T Cx(t) + 2~ T (y(t), t)D T Cx(t) ~ ~T ( y(t l ~t)D T D~(y(t), t) + -~-~2~ T (y(t), t)~(y(t), t)
=
-IIN- 89
+ D T C ) x ( t ) + N 89 1 T --{yT(t)y(t) - -s (y(t), t)~(y(t), t)}.
2 (50)
244
YASUHIKO MUTOH ET AL.
From eq. (48), this implies that
AV(x(t))
(51)
holds for any ~(y(t), t) in the sector [-k,k], and the conclusion follows. (Q.E.D.) Consider now a general sector [kL, ku](kL < kv). Let
ku + kL k0
k
~"
=
~
~
2
ku -- kL 2
>0.
Then, kL -- ko - k, ku - ko + k and
{~,(v(t), t) - k~v(t)} T {~,(y(t), t) - k~v(t)} = {(o(y(t),t)+ ky(t)}T{Co(y(t),t)- ky(t)} where ~(y(t),t) - ~o(y(t), t ) - koy(t). Thus, the quadratic constraint eq. (46) becomes
{~(y(t),t) + ky(t)}T{(o(y(t),t)- ky(t)} <_O.
(52)
Hence, the stability of system ~]M with the nonlinear part eq. (45) in the sector [--kL, ku] is equivalent to the stability of the following nonlinear feedback system. Linear part"
; ( t + a) u(t)
-
{ A - koB(I + koD)-lC}x(t) + B { I - ko(I + koD)-lD}fi (I + koD)-'Cx(t)+ (I + koD) -1Dft (53)
Nonlinear Part"
- -(o(y(t), t)
(54)
which lies in the sector I-k, k] (k > 0). Thus, from Theorem 4 the following sufficient conditions for the absolute stability of the system ~ M with respect to the sector [leL,ku]. T h e o r e m 5 System ~ M is absolutely stable with respect to the sector [/eL, ku] , if A is stable matrix and
I 1 0 ( ~ I - -e4)-1]~ ~- DIloo <
ku - kL
(55)
STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS
245
where
f~ [~ O D
- A - koB(I + koD)-lC -- B { I - ko(I + koD)-lD} -- (I+ koD)-lC - (I+ koD)-lD.
R e m a r k 1 For the case of a single-input and single-output system, Theorem 4 implies that if
Ic~(~J~z/- A)-~bl < f,1 Vw e [0,27r]
(a6)
then the system is absolutely stable with respect to the sector [-k, k], which corresponds to case 3 of Theorem 2 with --kL = ku = k. This can be interpreted as one of the circle criterions for the single-input and singleoutput system, because the H a norm condition ]lG(z)ll~ < 1 implies that all a-plots of G(e jw) lie in the disk with radius 1/k. From this point of view, Theorem 5 can be regarded as a generalized circle criterion for the multi-input multi-output case. | R e m a r k 2 Consider now only the sector described in eq. (46) where kL and ku are scalars for simplicity. However, when each feedback channel has different sector conditions, kL and ku can be replaced by the diagonal matrices KL and Ku, but this does not affect the argument of this section.
B. QUADRATIC STABILITY The absolute stability will now be discussed from a different point of view, one in which the nonlinear feedback part is regarded as a time-varying structured uncertainty of the linear part. This can be regarded as a robust stability problem. The concept of the quadratic stability is used for dealing with the stability of this kind of system. Consider a linear system with a time-varying perturbation described by
x(t + 1 ) - [A + AA(t)]x(t)
(57)
where AA(t) denotes an uncertain perturbation matrix which can be expressed by A A ( t ) - EF(t)G (58) with known matrices E and G with appropriate dimensions and unknown function F(t). AA(t)is said to be admissible if FT(t)F(t)vt.
246
YASUHIKO MUTOH ET AL.
D e f i n i t i o n 8 System eq. (57) is said to be quadratically stable if there exists a positive definite matrix P and a positive scalar a > 0 such that
xT(t){[A + AA(t)]Tp[A + A A ( t ) ] - P}x(t) < --axT(t)x(t), Vx(t) E R ~
(59) for all admissible AA(t) and t -
0, 1,2,---.
|
It is clear that if the condition in Definition 8 is satisfied, then the stability of system eq. (57) will be guaranteed by the Lyapunov function V(x) xT(t)Px(x) for all admissible perturbations. The following result is a slight modification of the results for continuous time systems discussed in the literature [11] [12]. T h e o r e m 6 Assume that A is a stable matrix. System eq. (57) is quadratically stable if there exists a scalar A > 0 and a positive definite matrix P > 0 such that N "- I - A2ETpE > 0 and the Riccati inequality 1 T G + A2AT P E N - P + A T PA + -A-fiG
1
ET PA < 0
(60)
holds.
II
(Proof) Suppose that there exists a scalar A > 0 and a positive definite matrix P > 0 satisfying I - A2E T P E > 0 and eq. (60). Let
Q "- - { - P + A T PA + ~-~GTG + ~ 2 A T p E N -1E TPA} > 0
(6a)
Using eq. (61) and some algebraic manipulations gives
[A + AA(t)] T P[A + A A ( t ) ] - P = +G T F T (t)E TPA + A T PEF(t)G + G T F T (t)E T P E F ( t ) G + - 1f i g T G + A2AT P E N - lET P A - Q =
-Q-
1 T ~-ffG { I - FT(t)F(t)}G
-{AATpE-
GTFT(t)N}N-I{AETpA-
-~NF(t)G}.
(62)
Hence,
xT(t){[A + AA(t)]Tp[A + A A ( t ) ] - P}x(t) < --~xT(t)x(t), Vx(t) E R n (63) for a sufficiently small positive scalar c~(< o',~in(Q)), where O'min(') denotes the minimal singular value.
STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS
247
Consider now a more general class of perturbations. Suppose that the perturbation F(t) in system eq. (57) satisfies
F T(t)F(t) <_ k2I, vt for a given k > 0. Let E -
(64)
kE a n d / ~ ( t ) - ~F(t). Then,
A A ( t ) - E F ( t ) G - EF(t)G and /~(t) is admissible. Corollary.
(65)
Thus, from Theorem 6, we have the following
C o r o l l a r y 2 For given k > 0, system eq. (57) with the perturbation satisfying eq. (64) is quadratically stable if there exist a scalar ~ > 0 and a positive definite matrix P > 0 such that k 2 I - A2ETpE > 0 and the Riccati inequality 1 T G + A2AT P E { k 2 I - ~2ET RE} -1 E TPA < 0 (66) - P + A TPA + ~-~G
holds.
!
From Lemma 3, it is easy to show that the condition in Corollary 2 is equivalent to the H ~ norm constraint IIC(zI - A) - Etl
<
1
(67)
with a stable matrix A. For the continuous time systems, it has been shown that this H ~ norm constraint is the sufficient and necessary condition for the system to be quadratically stable [11], [12]. Consider the stability of system eq. (57) from the absolute stability approach. System eq. (57) can be expressed as consisting of two parts. Linear Dynamic Part"
x(t + l)
-
Ax(t) + Eu
y(t)
-
C (t)
(68)
Feedback Part 9
u(t) - -~(y(t), t) - -F(t)y(t),
F T (t)F(t) < k2I.
(69)
It is clear that eq. (69) satisfies the sector constraint eq. (48). Hence, from Theorem 4 the system defined by eqs. (68) and (69) is absolutely stable if A is a stable matrix and
Ila(zI-A)-XEIIoo <
1
248
YASUHIKO MUTOH ET AL.
holds. It follows also from Corollary 2 from the point of view of the quadratic stability. It should be noted that, in general, quadratic stability is not equivalent to absolute stability, since quadratic stability requires the existence of a fixed quadratic Lyapunov function and not absolute stability.
C. EXPONENTIAL STABILITY When a system is stable, it is of interest to know if the state of the system is exponentially bounded. In the rest of this section, attention will be focused on the exponential stability. It will be shown that under the conditions given in Theorem 4 or Theorem 6 the system is exponentially stable. Consider a nonlinear system given by
x(t + 1 ) - f ( t , x ( t ) )
(70)
where f " R • R n ---, R n is C p (p >_ 1) function in z with f(t, 0) - 0, vt ~ R. D e f i n i t i o n 9 System eq. (70) is said to be exponentially stable if and only if for any to and xo - x(to), there exist constants 0 < fl < 1 and p(.~0) > 0 such that the solution of eq. (70) with z(to) - xo satisfies
IIx(t)l[ < p(~0)~ (~-'~
(71)
for all t >__to.
l
D e f i n i t i o n 10 A continuous function ff 9R + ~ function, if it is strictly increasing with ~(0) - 0.
R + is called a class-K; l
The following Lemma is a slight modification of Lemma 1 in [13]. L e m m a 4 For system eq. (70), if there exists a function Y ( t , x ( t ) ) with V(t, 0) - 0 such that
1) y(t,~(t))>_ r
vt _ to.
2) A V ( t , x ( t ) ) - V(t + 1,x(t + 1 ) ) - V ( t , x ( t ) ) < M - c r V ( t , x ( t ) ) ,
vt >_ to.
for some ff E/C, and constants M > 0 and 0 < a < 1, then t-to-1
r
< w(t, ~(t)) < (1 - ~)('-to)W(to, ~(t0)) + M Z i=0
(1 - a) i. (72)
STABILITY OF DISCRETE NONLINEAR FEEDBACK SYSTEMS
249
(Proof) From condition 2 ,
V(t,x(t)) <_ M + (1 - a ) V ( t -
1,x(t - 1)).
(73)
By using condition 2 again
V(t,x(t))
<_ M + ( 1 - a ) V ( t - 1, x ( t - 1)) _< M + M(1 - a) + (1 - a)~V(t - 2, x(t - 2)).
(74)
Hence, eq. (72) follows by repeating use of condition 2). T h e o r e m 7 Consider system eq. (57). If there exist a scalar ,k > 0 and a positive definite matrix P such that N
.!
I-
~2ETpE > 0
Q
:~
-{-P
(75)
1 GT G + A2AT P E N - 1E T PA} > 0 (76) + A T P A + --~
Then, for any admissible AA(t) system eq. (57) is exponentially stable with
C(xo)
"-
/ d
Xl
Pxo
(77)
v "-
x/'l - a
(78)
where a is any constant scalar satisfying (79)
(Proof) Suppose that there exist a scalar ,k > 0 and a positive definite matrix P > 0 satisfying eqs. (75) and (76). Let
r
m;
(P)llx(t)lt 2
(80)
where I1" II denotes the Euclidean norm. It is clear that ~ E/C. Define a function Y(x(t)) xT(t)Px(t). (81) -
Then, V(x) >_ ((]Ix[]) and from the proof of Theorem 6, AV(z(t)) __ -~mi~ (Q)x T (t)x(t)
(82)
250
YASUHIKOMUTOHET AL.
From eq. (82)
AV(x(t)) <_-a'U(x(t))
(83)
where a' - ~m.~(P)" ~m,.(Q) Hence, conditions 1) and 2) in Lemma 4 hold with M = 0 and a sufficiently small a > 0 satisfying -
o~ < rain{o/, 1}.
(84)
~)(t-t~
(85)
From Lemma 4, (([Ix(t)l[) _< ( 1 -
Substituting eq. (80)into eq. (85) gives
xToPxo [[x(t)[[ 2 < ~-~i:i-fi) (1 - a) (t-t~
(86)
which is the desired result. T h e o r e m 8 Consider system E M. If there exist a scalar A > 0 and a positive definite matrix P such that N
:=
I - A2ETpE > 0
Q
:=
-{-p+ATpA+~---~GTG+,~2ATpEN-1ETpA}
(87) > 0 (88)
then, the system with the sector I-k, k] is exponentially stable with
C(xo)
"-
i amin(P) Pxo
(89)
3
:=
vq-
(90)
c~
where c~ is any constant scalar satisfying
crmin(Q) a < mint ~"~E(P-)' 1}.
(91)
(Proof) Define ( ( [ I x ( t ) ] [ ) = ~min(P)llx(t)t[ 2 and V(x(t)) = xT(t)Px(t). As shown in the proof of Theorem 4, using the Riccati inequality eq. (85),
<
- x T ( t ) Q x ( t ) - ]]N-}(BTpA + DTc)x(t) + N}~(y(t),t)]] 2 i
-{ur(t)y(t) <
_
-
amin(Q._______~) V(x(t)). mo (P)
t)}
(92) (93)
Therefore, the desired result follows in the same manner as the proof of Theorem 7.
STABILITYOF DISCRETENONLINEARFEEDBACKSYSTEMS
251
R e m a r k 3 Theorem 7 implies that the quadratic stability of system eq. (57) is equivalent to the exponential stability. In other words, if the system is quadratically stable then for any given x0 the state trajectory of the system is exponentially bounded independently on AA(t). ! R e m a r k 4 Since the existence of such ,~ > 0 and P > 0 that satisfy eq. (87) and eq. (88) is equivalent to the gcr norm constrain LID + C ( z I - A ) -1 BJJoo < ~, it is clear that if the absolute stability of system ~]M with the sector [-k, k] is guaranteed by the condition in Theorem 4, then the system is exponentially stable and the convergence rate ~ does not depend on the nonlinear part 9(Y(t))|
VI. CONCLUSIONS The absolute stability problem of discrete-time nonlinear feedback control systems was considered in this chapter from three aspects. The ordinary circle criterion for single-input single-output systems was first considered. Attention was then focused on the simultaneous Lyapunov sector condition, under which the nonlinear feedback system is absolutely stable. It was shown that the simultaneous Lyapunov sector can be related to the linear dynamic part through two real roots of a certain algebraic quadratic equation. The absolute stability problem of multivariable nonlinear feedback systems was then considered. The H ~ - n o r m condition was applied to this problem, which can be regarded as a circle criterion for multivariable systems. In this discussion, the matrix Riccati inequality played a basic role. Finally, it was shown that the absolute stability guaranteed by the H ~ - n o r m condition implies exponential stability.
REFERENCES 1. J.L.Willems, Stability Theory of Dynamical Systems, Nelson,1970 2. M.Vidyasager, Nonlinear Systems Analysis, Prentice Hall, Englewood Cliff, N.J., 1978 3. A.Isidori, Nonlinear Control Systems, Springer-Verlag, New York, 1989 4. H.Nijmeijer and A.J.van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990 5. M.A.Aizerman and F.R.Gantmacher, Absolute Stability of Regulator Systems, Holden-Day, 1964
252
YASUHIKOMUTOHETAL.
6. R.E.Fitts, Two Counter Examples to Aizerman's Conjecture, IEEE Trans. Automatic Control, Vol.ll, No.3, pp.553-556, 1966 7. S.Boyd and Q.Yang, Structured and Simultaneous Lyapunov Function for System Stability Problems, Int. J. Control, Vol.49, No.6, pp.22152240, 1989 8. K.Zhou and P.P.Khargonekar, An Algebraic Riccati Equation Approach to H~ Optimization, Systems and Control Letters, 11, pp.8591, 1988 9. A.Stoorvogel, The Hoo Control Problem: a State Space Approach, Prentice Hall, 1992 10. K.Zhou and P.P.Khargonekar, An Algebraic Riccati Equation Approach to Hoo Optimization, Systems and Control Letters, 11, pp.8591, 1988 l l. I.P.Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems and Control Letters, 8, pp.351-357, 1987 12. P.P.Khargonekar, I.P.Petersen and K.Zhou, Robust stabilization of uncertain linear systems: quadratic stabilizability and Hoo control theory, IEEE Trans. Automatic Control, Vol.35, pp.356-361, 1991 13. V.C.Aiyken and H.M.Schwartz, On the exponential stability of discrete-time systems with application in observer design, IEEE Trans. Automatic Control, Vol.39, No.9, pp.1959-1962, 1994
CONTINUOUS
TIME
EQUATIONS
AND :
DISCRETE
REVIEW
TIME
AND
NEW
Ahmad J.A. De
Department
I .
LYAPUNOV DIRECTIONS
A. Mohammad Abreu-Garcia
of Electrical Engineering The University of Akron Akron, OH 44325-3904
INTRODUCTION
Since theory
its
reintroduction
continues
and design
in the
to play a vital
of control
systems.
1950's,
Lyapunov's
role
in the analysis
This
role
is evident
through a large number of direct and indirect applications in controls. Originally
Lyapunov
to test the stability Lyapunov functions tant tools tems, vide
the
eling.
Lyapunov for
functions
systems.
However,
and analysis
equations.
stability,
system's
Lyapunov
have lead to one of the most imporNot
but
properties
In particular,
robustness,
of dynamical
in the design
a test
dynamical
introduced
they
of control
only
sys-
do they pro-
also
define
most
relevant
to system mod-
controllability,
observability,
and optimality can be extracted from these
equations. CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
253
254
AHMAD A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
This
chapter
Lyapunov's on
some
system
presents
theory
new
and
and
its
novel
modeling.
is n o t e d
In
nique
has
that
senting
this
example
of the
Section
II
Lyapunov's
theory,
theorems
tion
also
sents
This
both
into
brief to
these
intuitive
major
analytical
interpretation
of
jor a p p l i c a t i o n s
of the L y a p u n o v equations.
equations
and
an example.
presents The
a continuization
in Sections VI and VIII II .
HISTORICAL
were
first
amount
1950's,
for
solv-
Section V of these
technique
references
ma-
as
are
given
Lyapunov
func-
respectively.
from their name,
introduced
work on the s t a b i l i t y late
these
BACKGROUND
As can be inferred tions
and
sec-
IV summarizes
for the application
conclusion
of
III which pre-
methods
Section
to
impor-
This
equations.
out new d i r e c t i o n s
and
equations.
and n u m e r i c a l
good
review
ing the L y a p u n o v points
a
sections.
equations,
is. f o l l o w e d by Section
pre-
followed
historical
Lyapunov
tech-
that
gives
can be
four
the
an
that
divided a
technique.
of this
chapter theory.
related
gives
equations.
this
in
equations
It is felt
of Lyapunov's
gives
tant
in
of
emphasis theory
Lyapunov
version
[1,2].
new d i r e c t i o n s is
this
a new c o n t i n u i z a t i o n
technique
chapter
review
with
of
particular,
in
take full advantage This
applications
a preliminary
appeared
historical
applications
are u t i l i z e d to devise It
a brief
they
by Lyapunov
of dynamical
have
of investigation.
been
in 1892
systems.
under
a
in his
Since the
considerable
It should be noted that much
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
of
the
results
the 1960's
in
[3-14,
Developments ers,
optimal
Riccati
Equation
[19,20],
control
dynamical
control
by
Moore
problem
control
important Lyapunov
results
in this
area
systems,
this
and
work
only the
addressed
THE
ORIGIN
OF
EQUATIONS
Consider =
Ax,
and define v(x(t))
:
THE
x(0)
=
respect
first
H-infinity
to
the
robust
of
case
The
the most Although
linear
is relevant
interested
Then
to time yields
to
reader
for more details.
FUNCTIONS
(zero input)
system
[23,24] (I)
v(x(t))
as (2)
x T(t)Px(t),
dimensions.
and
LYAPUNOV
function
where P is any symmetric proper
linear
others
x0,
a Lyapunov =
the
for both
LYAPUNOV
the autonomous
and
[22] .
references
THE
geometrical
summary
theory
oth-
Algebraic
is presented.
here.
is referred to the above
to
problems
the
in
direct
reduction
extensions a brief
developed
of
order
follows
has
the
by W o n h a m
which
nonlinear
A.
study
solution
its
among
Lyapunov's
through
in model
and model m a t c h i n g
In that
include,
systems
with
developed
therein] .
the
[21],
been
through
design
[15-18],
applications
introduced
have
applications
optimization
method,
of
field
and references and
parameter
properties
this
255
positive
definite
differentiating
matrix Eq.
with
(2) with
256
AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA
v(x(t))
=
xT(t)P~(t)
and substituting Eq. (x(t))
=
(I) into Eq.
x T(t) [PA
According
to
+ iT(t)Px(t) (3) gives
+ ATp]x(t) .
Lyapunov's
17,22,23,25,
and references
be
definite
negative
(3)
for
(4)
stability therein],
system
theory
~}(x(t))
(I)
to
[15has
be
to
stable.
This implies that PA where
+ ATp Q is
trix. or the
=
-Q
some
Lyapunov's
of the
in the analysis be
servability
positive
semi-definite
first
of the d i f f e r e n t i a l
the dynamics will
(5)
is referred to as Lyapunov's
method.
solution
0,
symmetric
This m e t h o d
second
As
<
method
equations
system which makes
and design of control seen
later,
Eq.
(5)
LYAPUNOV
EQUATIONS
was
independence
Lj (t) .
of
linearly independent
=
is non-singular.
i
AS
originally
Essentially,
Go (Li, Lj)
requires
describing
systems.
reduces
to the
ob-
Lyapunov equation upon the choice Q = cTc,
The Grammian ear
direct
it impractical
where C is the output matrix of the dynamic B.
ma-
any L i (t)
two
system.
GRAMMIANS
used to study the functions
and Lj (t)
if their Grammian,
are
lin-
L i(~) said
and to be
defined as
tf
Li (t-~) Lj (~) d~, 0
(6)
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
Similarly, Grammians =
the
controllability
of the d y n a m i c
Ax
and
257
observability
system
+ Bu
(7a)
y = Cx + Du,
(7b)
can be w r i t t e n Wc (t, to)
as
i
=
[26]
t
e A(t-~) B B T e AT(t-~) d~,
(8)
e AT(t-~) C T C e A (t-~) d~.
(9)
0
Wo (t0, t ) = 0
It is e a s y to show that then as to Lyapunov AWc
and
0 and
=
+ WoA--
are
equations test
(8) - (9) s a t i s f y
the
BBT ,
(i0)
_
cTc,
(11)
termed
respectively
play
Lyapunov
an
of c o n t r o l
systems.
for stability,
they
as
the
controllability
equations.
important
and c o n t r o l l a b i l i t y C.
--) oo, Eqs.
q
-
observability
design
t
matrix
equations
+ WcAT
ATWo which
-~
if A is a s t a b i l i t y
role
in
The the
In a d d i t i o n
also d e f i n e
latter
two
analysis
and
to p r o v i d i n g
a
the o b s e r v a b i l i t y
as can be seen next.
CONTROLLABILITY,
OBS ERVAB
I L I TY,
AND
MINIMALITY
In
this
section,
controllability
and
basic
definitions
observability,
and
and hence
tests
for
minimal-
258
AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA
ity,
via L y a p u n o v
play
a central
equations
role
are given.
in the
modeling
These and
concepts
design
prob-
lem.
Definitionat
t=to,
which
if
will
origin.
System one
can
pletely
state
the
is true
states,
then
it
then
if
controllability
A W c + Wc AT
t=t o
-
-BBT
tial
System
where
times
can
the
Test-
said
to
be
com-
If
system
controllable the
(7)
if and
unique
is only
solution
(12)
be
is
and
all
said
determined and
u(t)
t0
to
states
be
from
over
If this
initial
Test"
it is c o m p l e t e l y Grammian,
observable the
output
a period
is true then
state o b s e r v a b l e
Grammian
observability
Lyapunov
is
and all
[23,27].
Grammian,
(7)
y(t)
Observability then
(7)
times
the
q
said to be c o m p l e t e l y stable
x(t o ) to
[23,24] .
functions
[t0,tl],
state
for all initial
u(t)
equation
if x(t o)
input
controllable
initial
is c o m p l e t e l y
is n o n - s i n g u l a r
Definition-
be
function
system
stable
to
input
Grammian
to the L y a p u n o v
said an
controllable
Controllability the
is
construct
transfer
If this
initial
(7)
If
observable the unique
of
at and
time
for all ini-
system
(7)
is
[23]. system
(7)
if and only solution
is if
to the
equation
ATw O + WoA is n o n - s i n g u l a r
=
-cTc, [23,24].
(13)
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
Definition-
System
reducible number
if
the
needed
(7)
is
number
to
said
of
realize
to be m i n i m a l
states
the
259
n
is
the
corresponding
or
ir-
minimum transfer
function. It s h o u l d be n o t e d that, output zero not
(SISO)
case,
(MIMO) given
clear
in
case.
input
a s y s t e m will be reducible
cancellations as
in the single
occur.
the
However
multiple
A test
the
input
for m i n i m a l i t y
single if pole
situation
multiple for both
in the f o l l o w i n g t h e o r e m which
is
output
cases
is
is s t a t e d without
a proof. Theorem-
System
ducible
if
it
(7)
is
is
both
said
to
be
minimal
controllable
and
or
irre-
observable
[23,24] . Methods a given fer
for
reducible
function
rely
obtaining
on
are
available the
of the system.
Lyapunov
Stability
and
if
trix Q there
Test:
each
exists
+
PA
D .
INTERPRETATION
unobservable
(7)
is
stable
a b i l i t y concepts
if
symmetric
ma-
a positive
definite
symmetric
ma-
[23,24]
=-Q.
importance
all
definite
(14) AND
CONTROLLABILITY
The
and
They
positive
trix P which satisfies ATp
System
for
from the trans-
[20,23,24,28].
controllable
subspaces
for
realizations
one or even d i r e c t l y
isolating
only
irreducible
of
IMPORTANCE AND
the
OF
OBSERVABILITY
controllability
introduced earlier
in this
and
observ-
chapter
is
260
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
evident
in state
a powerful given tion
feedback.
technique
system.
to
In this
available
on the
the
system.
system
is m o d i f i e d
tions. the
In
If
the
states
If the
about
all
strategy,
is
the
system
the
the
is m a n i p u input of
the
specifica-
observable,
then
could be d e d u c e d
available
is
informa-
behavior
some d e s i r e d
states
of
the control
completely
and m a d e
feedback
response
of the
achieve
system
the
manner,
is
observer
[24].
this to
state
that m o d i f i e s
system
information
a state
reshape
control
lated via a c o n t r o l l e r to
Briefly,
via
to the c o n t r o l l e r
completely
controllable,
one
is able to design
a controller
that a r b i t r a r i l y places
the poles
system,
shaping
sponse
of
ability peated
thus
the
system
re-
[24] .
Theorems tions
the
on stability,
using
the
have been here.
Lyapunov
addressed
However,
the p h y s i c a l
controllability,
this,
suppose
that
state
x(0)=a,
the
functions/equations
earlier
and will
it is w o r t h w h i l e
interpretation system time
and observnot
is
response
SISO of
be
re-
investigating
of these equations. (I)
equa-
with
this
To do initial
system
is
given by h(tf)
=
eA(tf-to)a.
This means
(~5)
that the c o n t r o l l a b i l i t y G r a m m i a n
integral of hTh,
i.e
it
is a m e a s u r e
is the
of the e n e r g y
of
the time response. Another the impulse
interpretation
is
obtained
response of the system
by
considering
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
=
Ax
+
Bu,
x(0)
The impulse
response
h(tf)
eA(tf)B.
Thus
=
the
=
(16)
0.
of this
s y s t e m is given by (17)
controllability
Grammian
sents the e n e r g y of the impulse A
more
261
interesting
in this
repre-
response.
interpretation
c o n s i d e r i n g the f o l l o w i n g
case
is
obtained
by
o p t i m i z a t i o n problem:
Given the s y s t e m =
Ax
+
Bu,
it is d e s i r e d
x(0)
to d e s i g n
e n e r g y that d r i v e s final time t f > It has b e e n and o n l y (8)
=
if the
the
(~8)
x0,
a control states
input u with m i n i m u m
to the o r i g i n
at a g i v e n
to. shown
that
the
solution
controllability
is n o n - s i n g u l a r
[26] .
The
is p o s s i b l e
Grammian optimal
given
control
if
by Eq. law
is
given by U. where
-- B T e A T (t-t~
=
star d e n o t e s
demonstrates Grammian The
the optimal
the n e c e s s i t y
to be invertible,
condition
(19)
(to, tf) x0, control.
(19)
for the c o n t r o l l a b i l i t y i.e of full rank as a
for c o n t r o l l a b i l i t y .
observability
considering
the
Grammian
same a u t o n o m o u s
can
be
= Cx(t).
interpreted
s y s t e m with
taken as y(t)
Equation
(20)
by
the output
262
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
Now,
with
knowledge
interval state input Eq.
[t0, tf],
x 0.
of it
Without
is loss
of
both
[to,tf],
into Eq.
the
initial
the
control
for x(t)
from
(20) gives
of
Eq.
(21)
by
eAT(t-t~
T
__ e A T (t-to)c T C e A (t-to) X0,
Eq.
both
sides
re-
eAT(z-t~ CTy (~) dZ
x0 =
solution
to
can
given by
be
the
(22)
over
the
itf 0
interval
eAT(z-to)cTce A(~-t~ d~xo,
(~) d~,
necessity
for
the
order
to
invertible also
be
represents be
Eq.
eAT(~-t~
to the p r o b l e m
should
Grammian
=
Wo I (to, tf)
indicates
Grammian
of
(22)
(22) becomes
itf 0
This
Solving
the
in
integrating
It
find
over
(21)
sides
e AT(t-to) CTy (t)
which
to
output
C e A(t-t0) x0,
multiplying sults
and
generality,
zero.
(18) and s u b s t i t u t i n g =
input
desired
u can be c o n s i d e r e d
y(t)
or
the
seen
in
being noted
a measure by
(23)
(24) observability have
a unique
considered. that of
considering
the the the
observability output output
energy. energy
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
i
h
263
(25)
tf yT (~) y (I;) d~, 0
e A T (~-t0)c T c e A (~-t0) dZx0,
=
(26)
d to
=
x~W0 (to, tf) x0.
E.
THE
DISCRETE
Consider x(k+l)
The
=
x(0)
Lyapunov
time
EQUATIONS system
[15] (27)
= x 0, function
x (k) T p x (k) .
change =
V(k+l)
AV
=
x(k)T(ATpA
means
stable,
ATpA can
as (29)
-V(k),
that
the
or if one
(28)
in V is d e f i n e d
AV
This
one
the
LYAPUNOV
discrete
= Ax(k),
and define V
the
(27)
_p)x(k)
for
change
.
system
(27)
in V m u s t
be
(30) to
be
asymptotically
non-positive
definite,
writes -
p
=
derive
tinuous
time
meaning
of
-
(31)
Q,
similar
case.
stability
The
arguments only
in b o t h
to
those
difference cases
[24].
of
the
is t h a t
con-
of the
264
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
Except and
for
theorems
equations case.
are
similar
_
Wc
ATWo A
-
W o =_
time
-- -
of the
to
equations
AWcAT
the
meaning
concerning
These
where
stability, discrete
those
of
definitions
time
the
Lyapunov
continuous
time
are
BBT,
(32)
cTc,
(33)
matrices
[A,B,C]
are
those
of
the
discrete
system.
This
concludes
retical III.
aspects
the
Lyapunov
the
OF
THE
early
equation
LYAPUNOV
1960 's,
This
importance
and
review
of
the
theo-
functions/equations.
has b e e n
investigation.
extreme
historical
of L y a p u n o v
SOLUTIONS
Since of
the
the
under
is
EQUATIONS
solution
a considerable
particularly
vast
to
due
the
amount
to
their
number
of
applications
took
on
three
in
the area of controls. The
research
in
this
area
directions.
Namely,
tion
implications,
and
tions,
its
and n u m e r i c a l
In the solution, [29],
one
finds
Barnett
Barnett [16].
in this
Section
II.
et
aspects
closed
the
solu-
form e x p l i c i t
solu-
the et
key
al
al
theoretical papers
[30] . [15],
direction the
have
by
aspects
Gantmacker that most
already
interested
of
Taussky
Other m a i n
It s h o u l d be n o t e d However
of
solutions.
direction,
and O s t r o w s k i
include sults
first
theoretical
different
been reader
et
the al
references [31],
and
of the
re-
addressed
in
is r e f e r r e d
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
to
references
above
and
references
therein
265
for
more
details. In the the
second
research
when
the
Jordan gives
is
focused
system
form
in this
direction, is
or any
direction
a summary
continuous Other [34] .
It
require
in
other
be
special
form
Power
time
are
that
transformation
of
This
is
a
solution such
[32,33].
to
as
A key p a p e r Power
for both
Lyapunov
due
noted
form.
form.
form s o l u t i o n s
discrete
solutions,
the
to H.M.
contributions
the
a
form
finding
canonical
is due
should
canonical
given
and
closed
around
of s p e c i a l
time
major
the
equations.
Peter
Lancaster
explicit the
the
solutions
system
numerically
to
a
demanding
procedure. In
the
third
finds
two
main
where
the
error
direction, approachesin
imized.
For
referred
to
al
Bartels
[35],
based This
on
details the
widely A.
work et
reducing
approach
Bartels
and
the
is
methods
this
of
K.
HIGHLIGHTS
this for
presented.
first
Zietak
A matrix most
is
one
iterative
successively
min-
the
reader
is
[11-14],
Peters
et
approach
is
The
second
to
a lower
This
here
THE
one
reliable
[36].
OF
is
solutions,
approach,
[36] .
u s e d and is g i v e n
LYAPUNOV
In
of
the
Stewart
The
solution
al
the
numerical
Schur
and
approach
is is
form.
due the
to
most
in detail.
SOLUTION
OF
THE
EQUATIONS
section
a
detailed
the
solution
The
first
of the
method,
due
discussion Lyapunov
of
three
equations
to B a r n e t t
is
and Story
266
AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA
[15],
is
method
based
is due
on
the
to B a r t e l s
m e t h o d is due to S.J.
1.
As
Solution via
was
Barnett
This
using
the L y a p u n o v
+ ATp
=
The
[36].
second
The
third
[3] .
Equations
Products
earlier,
to s t u d y the t h e o r e t i c a l
PA
Stewart
Lyapunov
Kronecker
Story.
Consider
and
product.
Hammarling
of
mentioned
and
Kronecker
this
method
is
aspects
method
is
normally
due
utilized
of the solution.
equation
-Q,
Kronecker
to
(34)
products,
this
equation
can be w r i t t e n
as [(AT| where
+
vec(A)
transpose
(I|
T) ] v e c ( P )
defines
of the
A|
=
the v e c t o r
rows of A,
a11B a12B
--"
-
vec(Q),
f o r m e d by
(35)
stacking
alnB ,
......
the
and
. amlB
=
(36)
amnB
a s s u m i n g A is mxn. Equation A has
no
solution vec(P)
(35)
has
eigenvalues
a unique on
the
solution imaginary
if and axis
only
[15],
if the
is g i v e n by =
-
[ (AT|
+
(I|
T) ] - i v e c ( Q ) .
(37)
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
It
should
inversion
be
noted
that
of an n2xn 2 matrix.
is not u s e d for actual 2.
This
Bartels
is
one
and
of
popular MATLAB
AP
Stewart
the
most
is why this m e t h o d
Algorithm
numerically
For e x a m p l e
the
more
=
reliable
it is u s e d
and
in the
general
case
of
the
Lyapunov
-Q,
(38)
Q is symmetric.
lution
the
lets c o n s i d e r the e q u a t i o n
+ PB
where
This
involves
s o f t w a r e package.
consider
equation,
solution
calculations 9
widely used algorithms. To
the
267
if and
only
Equation
(38) has
if any e i g e n v a l u e s
a unique
[~i of
A
so-
and
~j
of B s a t i s f y (Xi + To
~j ~
solve
0 Eq.
for (38)
g e s t e d the f o l l o w i n g i)
Reduce
the
all
=
uTAu
=
and
for
P,
steps
9
matrix
unitary transformation
A
i
to
a
j.
(39)
Bartels
lower
and
Schur
Stewart
form
sug-
using
a
U as follows
Azl
0
.--
0
A21
A22
".
.
_ Apl
Ap2
---
App
where each m a t r i x Aii is at most
,
_
2x2.
(40)
268
2)
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
In
Schur
B
same
form
as
=
where 3)
the
V
The
vTBv
is
fashion,
B
B11
BI2
.--
Blq
0
B22
.--
B2q
0
-..
0
Bqq
=
also
reduce
a unitary
transformed
Q and
to
upper
triangular
(41)
_
matrix 9 P are
given
by
N
Q11 Q
=
uTQv
......
Q1q (42)
=
_
QpI
......
Qpq
P11
......
Plq
_
and
P
=
uTpv
(43)
=
Ppl 4)
Recursively,
(k=l,2,...,p; It
should
be
solve
...... for
the
Ppq _ blocks
of P
as
k-i
L-I
j=l
i=l
follows
1= 1,2,...,q). noted
that
the
(44) solution
of
Eq.
(44)
is
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
easy to carry mension
out
of 2x2.
since One
ber of c o m p u t a t i o n s B=A T.
This
Schur
should
a m a x i m u m di-
also notice
that
is r e d u c e d by almost
the num-
one half when
is needed.
of
the
computational
decomposition
(2+4g) n 3 flops,
and
where
tions
for
The actual
the
is one
due to
addition number
form
reduction
of Eq.
to
be
average
Schur
solution
is
estimated
a flop
and g is the
required
burden
is
multiplication, verge.
have
is due to the fact that only one Schur de-
composition Most
all blocks
269
the
about
and
of
one
iterato
(44) requires
conabout
7n 3 flops.
3. S.J.
Hammarling Hammarling
Algorithm
1982,
modified
the
Bartels/Stewart
a l g o r i t h m such that the solution to the L y a p u n o v equations
is
Hammarling directly
more was
numerically
able
for the
to
upper
solve
well the
conditioned.
Lyapunov
triangular
Cholesky
equations factors
of
the Grammians.
B.
SOLUTION
OF
THE
DISCRETE
LYAPUNOV
EQUATIONS As for
in the
the
possible system. reader
continuous
discrete
time
for specific For
details
time
Lyapunov
canonical of
is r e f e r r e d to N.J.
and references
case,
this
equations
solutions are
representations type
Young
t h e r e i n such as
explicit
of
of the
solutions,
[9], K e q q i a n WU
[38].
only the [37],
270
AHMAD A. MOHAMMAD AND J. A. DE A B R E U - G A R C I A
It s h o u l d be n o t e d that discrete ment
time
The
discrete
common
time
continuous rithms
procedure
efficient
practice
Lyapunov
time
can
nature
Lyapunov equation prevented
of n u m e r i c a l l y
tion.
the n o n l i n e a r
be
to
the d e v e l o p for the solu-
in n u m e r i c a l l y
equation
one w h e r e used
algorithms
solving
is to convert
efficient obtain
of the
the
it to a
and r e l i a b l e
algo-
solution.
This
the
is o u t l i n e d next.
Consider
the
discrete
Lyapunov
equation
given
by
[32,33] ATLA
-
L
=
- Q,
(45)
u s i n g the t r a n s f o r m a t i o n A
=
(B+I) (B-I)
converts
the
continuous BLb
+
discrete
LbB
=
(A-I)
it is
conditioned. for
the
tance.
=
(B_I)T
time
Lb 2 (B-I) ,
Lyapunov
(46)
equation
to
the
(47)
(47)
- Q. can
be
or
S.J.
Bartels/Stewart of
L
time L y a p u n o v e q u a t i o n
Equation However,
-I ,
should
be
solution
of
for
out
demanding
the d e v e l o p m e n t this
Lb
using
H a m m a r l i n g 's
pointed
numerically Thus,
solved
problem
that and
either
algorithm.
the
inversion
might
be
ill
of a new t e c h n i q u e is
of
great
impor-
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
C.
A
NOTE
ON
THE
DESCRIPTOR
SOLUTION
OF
Ex
= Ax
y =
EQUATIONS
system
+ Bu
(48a)
Cx,
where
(48b) -I
(sE-A)
condition The
THE
LYAPUNOV-LIKE
Consider the regular d e s c r i p t o r
exist
as
a
sufficient
and
necessary
for regularity.
expected
Grammians
controllability
for this
and
system satisfy
observability
[39]
A W c ET
+ E W c AT =
- BB T
(49)
ATWo E
+ ETWo A
- C TC.
(50)
Under
the
=
assumption
consistent
initial
eliminate
any
that
the
conditions
system
is
(initial
impulsive
behavior),
given theorems
concerning
reachability,
and
similar
stability
However,
the
unique.
In addition,
ness
with
of the
Grammians to
the
corresponding
divided to
these 1987
about
part the
of
with
conditions
that
Lewis
regular
systems.
equations [40],
the
is
not
has shown that
in general.
existence he has
two parts, the
noncausal
system part
has
observability,
He also
for these Grammians
Briefly, into
regular
Frank
for
satisfied
solution. causal
of
definition
theorems
are
those
Bender
are not
gave a more general gether
to
solution
these equations
ing
271
and
shown one
uniquethat
the
correspond-
and of
to-
the
the
other
system.
272
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
According
to
Bender,
the
teachability
Grammians
are
given by
r (AWc~ E T + ~.wo~A ~) r r
= - CoBB Tr T
(~.wo2~. ~ - Aw~2A =) r
while the o b s e r v a b i l i t y
= r162
are e x t r e m e l y applications tions.
izations, technique solution
and
the
one
hence,
to obtain to this
(54)
the coefficients
series expansion solutions This
require
For e x a m p l e
(53)
= CT-~CTCr
difficult. that
Tcr
respectively,
s o ,s -I in the Laurent
that
are given by
= - r
CT_~(~.TWo2~. - ;JWo2A) r
It is clear
(52)
Grammians
r (;JWo~. + ~.TWo~A)r
where ~-I and ~0 are,
(51)
the
can
specific
for these
solution
not
reduced
of H(s).
is a m a j o r
can not
of
obtain utilize
equations
drawback
in
of these
equa-
balanced
real-
the
order models.
balancing However,
p r o b l e m has been p r o p o s e d
a in
[2].
This
concludes
the
review
of
the
solutions
to
the
Lyapunov equations. IV.
IMPORTANT
APPLICATIONS
LYAPUNOV
Different have
already
OF
THE
the
Lyapunov
EQUATIONS
applications been
of
introduced
However,
it is
felt
that
of these
applications
equations
in the p r e v i o u s
a more
is n e e d e d
comprehensive for the
sake
section. summary of clar-
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
ity and completeness
of this
of these applications A.
LYAPUNOV
These
Consider =
theory
represent
and
a summary
here.
+ Bu,
the
in control stable
x(0)
=
first
application
system design
[41].
system given by (55 .a)
x0,
= Cx.
(55.b)
It is desired to design uTu
Thus,
CONTROLLERS
the minimal
Ax
y(t)
is p r e s e n t e d
controllers
of Lyapunov's
chapter.
273
a control
input u such that
___ 1
such
(56)
that
the
initial
state
returns
problem
starts
to the
origin
as rapidly as possible. The
solution
Lyapunov V
xTpx
=
x T[PA that
symmetric unique PA
+ ATp]x A
is
symmetric
Substituting
The
by
assuming
a
(57)
=
control
2uTBTpx. which
(58)
implies
semi-definite
positive
definite
Q,
that
there
for
any
exists
(59)
(59) into Eq. +
input
a
P such that
-Q.
Eq.
- xTQx
+
stable
positive
+ ATp
=
this
function
=
Notice
to
(58) yields
2uTBTpx. u
should
(60) be
chosen
.such
that
V
is
274
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
negative
with
the
largest
magnitude
clear that u must be parallel two
vectors
must
have
ond term negative.
to BTpx.
opposite
signs
It
However, to make
It
is
the
sec-
Thus u should be taken as
known
(61)
important
optimal LQR
is
these
BTpx
u
the
possible.
to
notice
controller
design
that
among
gives
only
this
all the
controller
controllers. linear
is The
optimal
con-
troller. B.
LYAPUNOV
EQUATIONS
ANALYS
This
has already
beginning
of
time
descriptor
earlier. suffer
It
especially responding transfer
chapter.
was
some
the
that
to
at the
discrete
also
addressed
these
extensions
problems.
descriptor
case
into two
This
is
where
the
subsystems
and p o l y n o m i a l
parts
cor-
of the
function.
LYAPUNOV
EQUATIONS OF
Controllability, ity analysis and
the
proper
MINIMALITY
have
out
to be s e p a r a t e d
to
addressed
were
serious in
been
Extensions
systems
pointed
evident
system needs
C.
this
from
STABILITY
IS
application
and
IN
already
implications
THE
systems
addressed of
these
THE
SYSTEM
observability,
of dynamic been
DEFINE
and
hence
via Lyapunov
earlier.
concepts
The in
minimalequations
importance
control
system
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
design the
were also
considered
applications
sign,
observer
lems
have
noted
design,
been
that
of these
concepts
and
discussed
the
in details
of
controller
However,
these
systems
suffers
problems.
mainly
arise
tion
and
latter
the
two
problems
solution
cases.
of
Lyapunov
These
problems
de-
probit
concepts
crete time and d e s c r i p t o r These
Among
realization
in detail.
application
therein.
the
minimal
275
was
in dis-
from serious
in the
defini-
equations have
in
been
the
briefly
introduced earlier. D .
LYAPUNOV
EQUATIONS
IN
MODEL
ORDER
REDUCTION
B.C. the
Moore
[21]
demonstrated
controllability
'balanced was
used
der m o d e l s Lyapunov
constitute
and o b s e r v a b i l i t y in the
same paper
of d y n a m i c a l
equations
and is essential tion
observability
coordinates,
controllability fact
and
algorithm
how the eigenvalues a measure
to obtain This
introduced
later;
the This
reduced
application
the most
in the d e v e l o p m e n t
in
of
of each state.
systems.
is p r o b a b l y
Grammians,
of
important
orof one
of the continuizahence
it
is
intro-
duced here in detail. 1.
Since
The
Technique
its i n t r o d u c t i o n
technique
has
[42-53].
By
niques
Balancing
triggered now,
(BT)
by B.C. Moore, an
there
to obtain b a l a n c e d
intensive
are
several
realizations
the b a l a n c i n g
wave
of
research
numerical
tech-
[see references
276
AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA
above] . rather
However,
than the
they
only
concept.
differ
An o u t l i n e
in
the
of this
details technique
is g i v e n next. Consider
y
the
system
[50,52,53]
=
Ax
+
Bu
(62a)
=
Cx
+
Du,
(62b)
and the c o r r e s p o n d i n g AWc
+ Wc AT
ATw o + Next,
consider
Wc = where
WoA
Lyapunov
equations
=
-BBT
(63)
=
-cTc.
(64)
the s i n g u l a r
value decomposition
of Wc
U c ~ c U T,
Uc
is
(65)
unitary,
~c
is
diagonal
with
entries
(;i~(Yi+ 1 - Let TI and
=
_
~i/2
UcLc
apply
a
,
(66)
similarity
transformation
to
system
(62)
to get
-1/2
A1
=
Zc
CI
=
CUcLc
Wcl
=
The next
Wol
I,
,.,1/2
.T..
Uc~UcZ~c
Wol
=
T 2 as
=
-1/2
Zc
1/2 T..... .., 1/2
~c
UcWoUc~c
step is to p e r f o r m
= UoiZoiUoT1-
Choosing
, B1
uTB, (67)
.
(68)
an SVD on Wol as (69)
277
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
= Uo Zo /' and
applying
system
(67)
a
(70) similarity
gives
transformation
the b a l a n c e d
using
T 2 to
system
_i14 T
Ab
-
/~oi U o l A I U o l ~ o I14 1 ,
Bb
=
~I/4UolB~ ir 2~oi
Cb
(71a)
_-I14
CiUol2~ol
=
,
(71b)
with the new G r a m m i a n s Wcb
=
_I14 T i14 Lol U o l l U o l ~ c
Wob
=
Lol
It mal
_-I14
should
and
-iI~
uTIWoIUoIEol
be
output
tained
from
_I12 ~oI =
=
noted normal
Eq.
i/2 Eol =
=
that,
the
by
E,
well
realizations
(71)
(72)
~, (~i>_(~i+l .
known
can be
applying
the
(73)
input
nor-
readily
ob-
similarity
transformations Ti
T 2 T IE ~/2
=
for input To =
normal,
nique
and
T 2 T I ~ -I/2
for output The
(74)
normal.
final is
(75)
to
step
in this
look
for
model
a break
order in
the
reduction singular
techvalues
such that (~r>>(~r + I ' and t r u n c a t e
the b a l a n c e d
system
after the
r th state.
278
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
To
illustrate
system
the
last
step,
:[A11 A12][xl
x2
A21
A22
bl
the
balanced
u (t)
b2
x2
~: [c,c.][x.]
consider
(76a) (76b)
X2
with
the G r a m m i a n s
Wc
then
=
the
'
~2
reduced
Xr
=
AllXr
Yr
=
CllXr,
with
I (Z~)>>k (Z2), min max
r th o r d e r m o d e l
(77)
can be t a k e n
as
+ bllU
(78a) (78b)
Granunians
Wcr For
o]
Wo =
-
Wor
=
simplicity,
renamed
as
2.
-- ~'r
the
(79)
reduced
order
model
matrices
are
[A r,B r,C r] .
Properties
of
the
Technique
a)
The
entries
of
the
system.
between
~i
them
t r i x H of the
of ~
are
There
a n d the system-
Balancing
[44, 4 5 , 5 2 ]
called is
an
singular
the
second
interesting values
Specifically,
order
modes
relationship
of the
Hankel
ma-
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
279
2 k i ( H H T) = ~ib)
In general,
and the
the
eigenvalues
controllability
of the
level
of
of
the
Grammians
controllability
are
and
observability a quantization
observability
of
each state. c) In balanced coordinates, as it is observable d) The subsystems
a state is as controllable
(Wo=Wc=~).
[Aii,Bi,C i] in Eq.
(76) are balanced
with Woi = Wci = ~i-
e) T r a c e ( W o o ) = ]~ D C 2
gain
f)
bound
Wco = C r o s s There
is
an
of
the
system,
Grammian. upper
on
the
frequency
error
given by ]E ( s ) ~ C g)
The
(s I-A) -IB-c I (s I-All )-IB iI~-<2 T r a c e (~2).
sub-matrices
Aii
are
asymptotically
stable
if
~i, ~2 have no common entries. Again, time
extensions
and
o f this
descriptor
drawbacks
that
application
systems
prevent
or
suffer
limit
its
to discrete
from
the
same
application
to
these systems. E.
LYAPUNOV AND
Briefly, use norms
the
ROBUST
basic going
system
analysis into
IN
H-INFINITY
CONTROL
the H-infinity
in the
Without
EQUATIONS
and robust control problems
no rms
IHL,
and design
details,
the
I~2,
I~F,
of control solution
and
~L~2
systems. to
these
280
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
p r o b l e m s boils down to the solution of the known algebraic For
Riccati
more
equation
details,
paper by G l o v e r Simply, tem
finity hand, a
internally
norm
of
Usually, inal
the
the robust
controller
Although chapter,
stable
control
the
key
a norm
consists
of
the c l o s e d loop sysminimizing
[22,54,55].
On
p r o b l e m consists an
the
the
in-
other
of d e s i g n i n g
uncertain
system.
system is given by a known nombound
on
the
fluctuations
from
system. these
it
is
topics
observed
are that
systems
and d i f f i c u l t i e s
in
LYAPUNOV
not
considered
extensions
to
in
this
discrete
suffer from the same drawthe
lution of L y a p u n o v equations F.
to
problem
while
stabilizes
time and d e s c r i p t o r backs
equations.
referred
control
system
that
and
the nominal
is
Lyapunov
that makes
the u n c e r t a i n
plant
reader
H-infinity
a controller
to be
the
[22] .
the
designing
the
and/or
EQUATIONS
definition
and
the
so-
for these systems. IN
SYSTEM
TRANSFORMATIONS Two ture.
important
These a p p l i c a t i o n s 1.
The bility tain
applications
Similarity
singular and/or
input
izations
are given next.
Transformations
value
decomposition
observability
normal,
introduced
are cited in the litera-
output
of the
Grammians normal,
earlier.
The
and
was
controllaused
balanced
importance
to
ob-
real-
of these
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
realizations
is evident
through
their
281
crucial
role
in
model order reduction. 2.
Continuization
Although directly of
a
the
suggests
that
investigated technique
on
the
its
This
resulting
is p r e s e n t e d
been
the
Grammians
discrete
used time
transformation,
direction
has
been
in a new c o n t i n u i z a t i o n
Lyapunov
next
not
equations.
This
details
serve
in more
to
for new Lyapunov based applications.
DIRECTIONS
LYAPUNOV A.
[2]
and
of
bilinear
possibility.
as an example NEW
the
have
equality
system
using in
discretization
equations the
time
based
technique
IN
THE
APPLICATION
novel
application
OF
THEORY
INTRODUCTION
In
this
section
equations
is
directions
in
out.
area,
continuous when
V.
Lyapunov
in this
version
and
More
a
developed. utilizing
Furthermore, these
specifically,
of
Lyapunov some
equations
are
a new c o n t i n u i z a t i o n
new
pointed
algorithm
is presented. The p r o p o s e d tinuization tions
and,
tinuization
continuization
scheme hence,
based is
a numerically
reliable
to
used
the
the
widely
LBCT
eliminates
on the use
termed
technique'
technique 'the
(LBCT). and
bilinear
less
of L y a p u n o v
Lyapunov
This
expensive
equa-
based
technique
transform.
the b i l i n e a r
is a new concon-
offers
substitute
Furthermore,
transform
drawbacks,
282
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
especially, tial
the
numerical
transient
error
Moreover,
this
the
widely
other
It is also the
zero
cial of
this
tween
Lyapunov
In this
and
that new
viewed
this
it
section,
approach
continuization
system
(CTS).
(LBCT)
time
that
system
(DTS)
The p r o p o s e d
guarantees
provided
advantage be-
modeling. may
lead
and
dis-
one
into a continuous
continuization
tech-
the s t a b i l i t y of the resulting starts
with
a stable
DTS.
The
is b a s e d on the assumption that both the CTS
B and output
dinates,
knowledge
Lyapunov
spe-
are u t i l i z e d to
It is shown that with a p p r o p r i a t e
the
and
PROBLEM
L y a p u n o v equations
input m a t r i x of
as
provides
system
to
hold.
transform
and the DTS should share the same Hankel ues.
order
Another main
theory
method.
Preliminary
a discrete
technique
can be
ini-
superior
zero
link
is h o p e d of
the
direct
CONTINUIZATION
1.
nique
the
this
the b i l i n e a r
technique. is
of
high
schemes b a s e d on Lyapunov theory.
THE
convert
technique, both
the
numerically
techniques
development
B.
is
stability
it
cretization
CTS
hold
approach
the
time
used
of this
In addition, to
technique
order
and
characteristic
shown that
cases
problems
and DTS,
one
equations
A.
Different
CTS
models
can
of
the
solve
for the
choices
matrix
choices
in b a l a n c e d
Hankel
the CT
C,
singular val-
two
singular continuous
system's
dynamic
of B and C result
corresponding
to
different
of the coorvalues time matrix
in d i f f e r e n t
discretization
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
techniques.
For
Euler's
method,
inverse
of
tion
of
while
Euler
be
that
the
the
of
ately.
As
B
in
all
this
tion
original
comes
the
more
CTS
resulting the
accurate
will
as
are
methods), trans-
normally been
dynamics DTS
chosen
and
in
to the
bilinear
CTS
as
a combina-
method
ob-
used. matrix
provided appropri-
discretization
give
This
the
is
model
only
system.
it
original
matrices
method
be close
in the
transform of
chosen
transform
continuization
techniques, of
C as
in the
those C
will
bilinear
the
and
CTS
(actually
bilinear
precisely CTS
the
result
eigenvectors
will
method
of B and
will
had
if B and C are
resulting
and
choice
form m e t h o d The
the
Euler's
the
tained
example,
283
an
approxima-
approximation
sampling
time
T
be-
becomes
smaller. to
In c o n t r o l
theory,
it is e s s e n t i a l
that
one be able
transform
a
CTS
and
vice
For
example,
CTS
model
given
in
and
simulation
is
asked
computer.
To
discretized
using
techniques In
the
asked
to
achieve
to
construct
for CTS
techniques
simulate
are
and
the
state
one
While
is
not
first
usually and
is
discretization understood,
[57,58].
a unique of
is
system
and well
nonlinearity space
a
discretization
model.
of
given
a digital
model
of the
lack
versa.
[56] .
implemented
the
CTS
problems,
are
is
it u s i n g
available
method
techniques is
one
behavior
a CTS
easily
this in
time
the
the
identification
continuization DTS
of
a DTS
problems,
this,
one
discrete
techniques reason
to
such as E u l e r ' s
system
given
into
The
main
mapping
from
identification
representations
[59] .
In
284
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
the p r o p o s e d technique, taining
a
CTS
lined.
This
method
servability among of
other
the
model
from
CTS
stable.
given
the
both
provided
that
Theoretically,
However,
of
equations
for
matrix
and
C,
emphasize tion of
a given
time
that this m e t h o d The
solution
of
to
out-
and
ob-
guarantees,
and m i n i m a l i t y
the
given
there
satisfy input
T.
will
both
matrix
It
DTS be
is in-
is
Lyapunov B,
output
important
to
can be used for discretiza-
only p r o b l e m
due
is
to a given DTS model.
will
set
sampling
as well.
the
them
model
This
stability
corresponding
one
DTS
for ob-
controllability
finitely m a n y CTSs only
procedure
equations.
properties,
and
a
uses
Lyapunov
resulting
minimal
a systematic
the
is the n o n - u n i q u e n e s s
nonlinear
nature
of
the
introducing
the
discrete time L y a p u n o v equations. The
following
continuization retical
section,
algorithm
aspects.
starts without
Theoretical
by
discussing
aspects
its
theo-
follow the algo-
rithm outline.
2.
The
Continuization
Algorithm
Let the DTS be given by Xk+ 1 = A d X k + Yk = where
CdXk
+
subscript
It is d e s i r e d the
behavior
singular
of
values
tain degree
BdU k
(80a)
DdUk
(80b)
d denotes
discrete
time.
to find a CTS model the
given
the
Hankel
of the CTS be those of the DTS,
a cer-
of accuracy,
DTS.
that approximates
in terms
Letting of norms,
is guaran-
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
teed.
More
specifically,
either
of the
system
can be p r e c i s e l y
the
tems.
If the
original
stable
DTS
is
[HI2 or same
285
the
DC
gain
for b o t h
sys-
then
so will
be
its CTS c o u n t e r p a r t . It can
should
use
the
lined b e l o w The with
be
for stable
solution
to
transform
in the worst technique nique.
More
without
the
available DT
recent
need
ATbZAdb where
~
subscript
-- Z
arise
that
=
-
out-
balanced The pro-
the
DTS
the
Therefore,
balance
a
DTS
transform
are
alleviate to the
the
use
developed
[2] .
DTS
be
will
on the
t r a n s f o r m tech-
bilinear to
be
f r o m the p r o p o s e d
to
via
the
DT
solution of
Thus,
assumed
of
Euler's without in
bal-
(Adb, Bdb, Cdbr Dd, T) .
s a t i s f y the DTS L y a p u n o v e q u a t i o n s BdbB~b T
=
-- CdbC db
db stands the
use
the
into
method.
equations
is d i a g o n a l
Denoting
will
and g i v e n by
=
one
starts
of a DTS m a y be b a s e d
approach
s y s t e m must
Z
DTS
requires
techniques
recently
anced coordinates
-
problem
is due here.
a novel
generality,
AdNZ,A~N
given
Moreover,
problem,
m e t h o d has been
This
to
DTS's
procedure
and f r o m the b i l i n e a r
Lyapunov
of
scheme
errors
[60] .
balancing
the
continuization
case,
itself
the
of c a u t i o n
The b a l a n c i n g
bilinear
in
continuization of
continuization
balanced.
anti-stable
system
the
A note
for
systems.
a transformation
posed
loss
that
reciprocal
coordinates.
the
noted
with
-
Q,
entries
for d i s c r e t e
sampling
time
(81)
p
=
T,
(82) (~i_>(~j i f
i>j.
and b a l a n c e d the
input
The
system.
and output
286
AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA
matrices
of the CTS Bcb,
Bcb =
Bdbl~,
This
choice
Euler's
method,
trices CTS
except
Ccb can be taken as
Ccb = is
(83)
Cdb/~.
justified
for example,
if
one
does
not c h a n g e
for the m u l t i p l i c a t i o n
dynamics
matrix
Acb
is then
notices
these ma-
factor
chosen
to
that
~.
The
satisfy
the
CTS L y a p u n o v e q u a t i o n s
so
Acb%
+
~ATb
=
-- P / T - -
ATb ~
+ ~Acb
=
-
that
the
DTS.
method
and the
and
output
while
in
bilinear
the
noticing
or Eq.
transform one
same
of
the
combination
of
t r a n s f o r m method-
the
is
are
Grammians a
taken
as
values
method.
It
is
Acb
separately,
the
solution
if
one
uniqueness
of
there are two s o l u t i o n s Then
solves the
from
for
the s o l u t i o n
Euler's taken
as
worthwhile either will
Acb
Eq.
not
from
be
Eqs.
is unique.
solution,
Acb I and Acb 2 that
it follows
in are
for
simultaneously,
(84)-(85).
the
solves
(84)-(85)
Eqs.
(85)
singular
However,
show
Qc,
bilinear
unique. To
(84)
strategy
Hankel
if
(85)
-
matrices
the
that
=
have
this
method
(84)
will
Clearly,
Euler's input
CTS
Q/T
Pc
suppose
that
satisfy both
that
Acbl%
+
ZAcTbl =
-- Pc
(86)
Acb2~
+
T ~Acb2
- Pc-
(87)
Subtracting
Eq.
=
(87) f r o m Eq.
(86) and l e t t i n g
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
Acb I - Acb 2 =
287
N
yields N]~
+
]~N T =
This e q u a t i o n
reduces
(~jnij + (Yinji Similarly, must
(88)
O.
-
f r o m the
which
+
NT~
=
observability
Lyapunov
equation,
reduces
0,
N
(90)
to
(~inij + nji(~j = Comparing
Eqs.
(~i~(~j i f
i~j,
nij
=
nji
and
is
seen
it
that
(91) , that
and
the
provided
only
solution
to
the
Acb 2.
solution
solution
of
of Eqs. these
two
(84)-(85),
[61]
9
Provided
i~j,
Acb can be w r i t t e n
(~i~(~j i f
- Pcii 2(Yi
it
equations
simple p a r a m e t e r i z a t i o n
acbii =
to
is
that Acb I =
the
that
that
-- 0,
implies
Returning
(91)
0.
(89)
these two e q u a t i o n s
seen
(89)
0.
satisfy
~N
which
to
is
is
a
as (92)
288
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
for the d i a g o n a l
elements
choice
that
the
of b a l a n c e d
Clearly, (Acb,
elements
system
Ccb,
as
This
is due
and e x p l a i n s
to the
coordinates.
Dc=Dd)
it
of Acb.
is b a l a n c e d
from L y a p u n o v
Bcb,
balanced
(93)
2 2 ((~i -- (~j)
for the off d i a g o n a l fact
and
Pcijf~j -- qcij(~i
acbij =
the
of Acb,
theory,
is
satisfies
both both
the
resulting
stable, Lyapunov
system
minimal Eqs.
and
(84)
and
(86). 3.
This
Theoretical
is best
Theorem-
Aspects
summarized
For
any
given
(Adb, Bdb, Cdb, Dd, ~, T) , modes,
there
balanced
exists
CTS model
Bob = B and
Acb
/fT,
is
a
of
the
LBCT
in the following stable,
with
minimal
distinct
unique,
theorem-
balanced second
minimal,
order
stable,
(Acb, Bcb, Ccb, Dc=Dd, ~),
DTS and
where
Ccb = C b/f{,
the
unique
solution
of
the
CT
Lyapunov
equations
AcbZ
+
~ATb
=
T - BcbBcb
=
- TBdbBTb
T Acb~
+ ~Acb
=
-
T CcbCcb
=
- TC~Cdb.
the
CTS
Moreover,
the f o l l o w i n g
approximation
properties-
of
the
DTS
model
has
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
i)
The
model 2)
CTS
model
is m i n i m a l
There
minimal
a
solution
if
the
DTS
for
Eqs.
(84)-(85),
for
a
of 2) is unique.
4) The I~2 n o r m of the DTS CTS
stable
of T, B, and C.
3) The s o l u t i o n the
and
and stable.
exists
fixed choice
is
289
(notice
that
we
error
in
is equal only
to the ~HI2 n o r m
deal
with
the
of
strictly
p r o p e r part) . 5)
The
initial
the
step
response
is
identi-
the
Hankel
cally zero.
Proof
:
(i) G u a r a n t e e d by L y a p u n o v equations. (2,3) (4)
Proved A
in s e c t i o n
direct
consequence
S i n g u l a r values (5)
First,
same value
it
p r o p e r part
4.
is
noted term
the
that
D.
is i d e n t i c a l l y
Comparison
with
matching
Euler's
both
systems
Moreover,
initial
of
share the
the
the
initial strictly
zero.
Other
Techniques
the p r o p o s e d
and
from
response
with
In order to v a l i d a t e parison
of
of the DT and CT systems.
feedthrough theorem,
2.
the
technique,
Bilinear
a com-
transform meth-
ods is p r e s e n t e d next. To model
set
the
stage,
suppose
and the c o n t i n u o u s
discretization In this
case,
technique the
first
model using
that
the
are r e l a t e d a
relation between
sampling the DTS
discrete
via Euler's period
T.
and the CTS
290
A H M A D A. M O H A M M A D AND J. A. DE AB R E U - G A R C I A
state
space m a t r i c e s
Adb =
I
+ TAcb,
Substituting (I
+
Eq.
Bdb =
(94)
TAcb)~(I
where
Wcb
will
be
anced' Thus, to
is
(95a)
BobBc b
(95b)
method,
must
satisfy
T Bcb Bcb,
-
that
Hankel
t e r m in Eqw. <<
(96)
necessarily
diagonal.
this having
for the CTS
same
TIAcb~AT~
T - TBcbBcb
=
from Euler's
its G r a m m i a n s
the
(94)
(81) gives
AcTb = -
later
in order
nonlinear
=
not
shown
with
have
+
- ~
Ccb~-
equation
T WcbAcb
AcbWcb+
, Crib =
into Eq.
as o b t a i n e d
the CT L y a p u n o v
Bcb~
+TATb)
TAcb AcTb + Acb The CTS,
is given by
system the
obtained
norm
as
However is
it
'semi-bal-
same eigenvalues. via Euler's
that
of
the
(95b) must be close
method
DTS,
the
to zero or
IBcbBT~
(97a)
or
TIAc << ~SdbS~/(~max. By comparison, LBCT This
has
the
the
same
s y s t e m must
system
Hankel
satisfy
(97b) obtained norm
as
via the p r o p o s e d that
the CT Lyapunov
of
the
DTS.
equations
Acb~
+
~cb
=-
B cbBTb,
(98)
~cb~
+
~ALb
T -- _ CcbCcb"
(99)
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
Thus
the
have
a
system
resulting
different
values
from
However
from the p r o p o s e d
dynamic
the
system
both
models
contrast
to
matrix
and
obtained
share
the
LBCT
Hankel
via
same
291
singular
inverse
input
will
Euler.
and
output
matrices. In
proposed put,
algorithm
bilinear
will
have
and output matrices
singular the
the
values.
proposed
system
algorithm via
different
while
Moreover,
obtained
transform
the
the
dynamics,
in-
sharing the
the
is
method,
system
strictly
bilinear
same Hankel
resulting
proper
from
while
transform
is
the
simply
proper. In
addition
to
for the previous important
providing
theorem,
(97b)
to
Of course
stability
Equation to
Euler's
be
gives
algorithm
tions
this
different
proof
following
the
in
Euler's
method.
by the theorem. use
and
the
This
for the p r o p o s e d
inverse of
Lyapunov
accuracy
eigen
less expensive
equa-
tests
analysis
with
stability/accuracy
line
of
investigation
for the test has
[2] . the
to notice
has
approach.
that Eq.
generalized
in the delta
equation
the
stability
(97).
used to define
to
suggests
It is i n t e r e s t i n g
and DTS
novel
have the
the condition
replacing
Eq.
been p u r s u e d 3)
the above
is g u a r a n t e e d
provide
method on
close
(97b)
computationally based
alternate
consequences-
I) Equation
2)
an
operator
been
Grammians approach
reached
In the
(95b)
for both
[62] .
independently
delta
has been
operator
CTS
However, using
a
approach,
292
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
this e q u a t i o n
is d e r i v e d on the basis
ergy control origin there
input that takes the initial
in a f i x e d time. is no
addition, ones.
equations
However,
tions
are
used
similarity surprise
assumption
to
between
since
operator
are
here
the
state to the
In the previous
optimality
in the delta
Lyapunov
of a m i n i m u m en-
the
from
original
derive
the
two
delta
start
approach,
obtained
the
to
development,
generalized
Lyapunov
generalized
approaches operator
In
the original
the
DTS
with.
equa-
ones.
The
s h o u l d be of no
approach
is
essen-
tially a v a r i a t i o n of Euler's method. 4)
Equation
assumption its
that
a
continuous
proaches to
(95b)
zero
Lyapunov
This
Eq.
when
a
to
system
the
the
logical
converges
sampling
seen by
This
into
proof
time
c o u l d be
(95b) .
equation
a solid
discrete
version
zero. in
gives
time
letting
transforms
continuous
the
one
to ap-
T equal discrete
giving
the
a f o r e m e n t i o n e d proof. 5)
Inspection
of Eq.
o b t a i n e d via Eqs. that
obtained
(95b)
(92)-(93)
via
the
reveals
that
the CTS model
is always more
inverse
Euler's
stable than
method.
can be seen by n o t i c i n g that the error term, t e r m in Eq. 6)
The
(95b),
previous
between
tion/ i d e n t i f i c a t i o n 5.
It the
Further
should
CTS
is
be
establishes
equations
and
a solid
system
link
simula-
concepts. Devel opment s
noted
equal
the first
is g r e a t e r than or equal to zero.
development
Lyapunov
This
to
that twice
the the
the
LCBT
steady trace
state of
gain
the
of
cross
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
Grammian
Wco
[21].
of
a matrix
is
it
is n o t e d
that
related
to
values
of
the
the
coordinates,
First of
is n o t e d
its
eigenvalues
Hankel
singular
following
and
the
that
the
eigenvalues.
the
the
Wco
sum
it
of W c o
293
trace
Secondly
are
directly
In
balanced
values.
r e l a t i o n b e t w e e n the e i g e n -
Hankel
singular
values
of
the
CTS holds : 2 This
can
be
exploited
to m a t c h
the
steady
state
gain
in the p r e c e d i n g
sec-
as follows: I) Solve
for the CTS as o u t l i n e d
tion. 2)
Find
DTS,
the
steady
state
gains
for
both
the
CTS
and
say sc and sd r e s p e c t i v e l y .
3) C o m p u t e the s c a l i n g 4)
Find
the
scaled
5)
Solve
for
the
s i n g u l a r values,
6.
factor set
CTS
of
using
f~=sd/sc. Hankel
the
singular
scaled
set
of
values
Hankel
as o u t l i n e d above 9
Numerical
Example
and
Further
Discussion It is w o r t h w h i l e tion t e c h n i q u e
comparing
with the well
U s i n g a M A T L A B program, of A p p e n d i x
the p r o p o s e d
continuiza-
known b i l i n e a r
transform.
a continuous
model
1 was f o u n d in three d i f f e r e n t
(i) The s t a n d a r d b i l i n e a r
t r a n s f o r m method.
for the DTS ways"
294
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
(ii)
Our
Hankel DTS,
proposed
singular
continuization
values
as
technique
obtained
by
using
the
balancing
the
and
(iii)
Our
scaled
proposed
set
of
continuization
the
in the p r e v i o u s
Hankel
technique
singular
values
using
as
a
obtained
section.
(iv) The Zero O r d e r H o l d method. The
results
i.
Different
are
compared
shown
of
responses with
in Figs.
As
for
the
step gave
is b a s i c a l l y
due
maximum
from
Hankel
is
est
maximum
ear
gave
state
where
arguments The tem
as
order
the
hold
In
is
for the
is
terms
of
the
impulse
obtained
via
the
the
proposed
the
uses
the
from
(iii)
as the
of
fact
error
the
via
relative
to that
to
system
bilinear
both methods
relatively
responses
and
the
as well
frequency hold,
method
via
This
while
obtained
errors
due
The
obtained
its m a x i m u m
signal
error.
error
this
that
are
transform,
term.
that
performance
has
initial
system
final
This
transform
to
results
bilinear
initial
that
it is c l e a r
superior
The
models
chapter.
obtained
The
error.
transform.
linear
a zero
(notice
and
the
in A p p e n d i x
continuous
feed-through
values
initial
(error/signal), (iii)
shown
DTS.
high
comparable
method) .
zero
above
relatively to the
singular
gave
of the
responses,
transform
transform
are
original
(ii) g a v e
error
bilinear
the
process
1-8 at the end of the
as expected, obtained
this
error (ii)
the
that
in the
small.
lowand
bilinthe
bi-
initial Similar
response.
of the c o n t i n u o u s
bilinear
transform,
algorithm
are
time
sys-
the
zero
shown
in
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
Figs.
5-8.
proposed all
low
This
It
is
clear
from
algorithm,
matches
frequencies
within
is
specially
true
these the
zero
the
for
figures, order
system's
the
system
295
that
the
hold
for
band-width. obtained
via
scaling. In tion
summary,
this
of L y a p u n o v
section
presented
equations.
This
a link b e t w e e n
continuization
niques
one
on
Lyapunov method els. same
the
on
the
for
CTS
model
or a s c a l e d given
the
will
two
the
for
DTS.
is
provided
theory
The
technique
provides
equations
steady hoped
a
have
mod-
either
the
singular
values
a scaling
factor
responses
that
this
technique
new
continuization
of the
of this
minimal the
input
and
technique
CTS
version
previous
parameterization balanced
a
step
choices
feature
DTS
of
state
developing
of
given
to
that
a stable
in
for
Hankel
shown
converse
tech-
stability
chosen
is
A main
that
provides
the
on d i f f e r e n t
The
application
and d i s c r e t i z a t i o n
of the
the
to p r o d u c e
a given
be
It
way
based
ability
Lyapunov
can
applica-
versions
It was
matrices.
true
CTS
to m a t c h
the
techniques output
hand.
systems.
pave
and
version
DTS.
can be c h o s e n of
other
obtaining
The
of the
hand
a new
is of
statement of
coordinates
the
DTS
can
be
of
the
found. The
choice
CTS p l a y s ity
of
equated
the
input
an i m p o r t a n t
this
technique.
to t h e i r DTS
justification method.
of
for
However,
role In
and
output
in the the
counterparts
this
choice
matrices
accuracy
foregoing
and v a l i d these
s c a l e d by i / ~ . was
it s h o u l d be b o r n e
based
on
were The
Euler's
in m i n d that
this
296
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
is not the only choice.
An alternate
choice c o u l d use
the b i l i n e a r t r a n s f o r m or the zero order hold method. VI
.
CONCLUSION
In
this
chapter,
a
background
material
for
presented. solution brief
historical
This
was
the
Lyapunov
of
the
Finally,
a new
system modeling
established. links
were
development equations between
Possible pointed of this
were
out.
equations
were
of
this
review,
theory
Lyapunov
a
pos-
were p o i n t e d out. based
on
in full details.
the New
and Lyapunov t h e o r y were In
to
the
exploit course
the G e n e r a l i z e d
these of
and
(97) which equations
system
simulation
lays the foundation in
the
the
Lyapunov
i n d e p e n d e n t l y d e r i v e d and a direct
Lyapunov
the
of the L y a p u n o v
technique
directions
technique
e s t a b l i s h e d via Eq. utilizing
Throughout
was p r e s e n t e d
a brief
Furthermore,
applications
continuization
and
a discussion
of the a p p l i c a t i o n s
Lyapunov equations links b e t w e e n
by
equations.
was presented.
sible e x t e n s i o n s
Lyapunov
followed
review of important
equations
review
analysis
link was for of
d i s c r e t i z a t i o n techniques. VII
.
REFERENCES
I.
A h m a d A. Mohammad, J.A. De A b r e u - G a r c i a , and Tom T. Hartley, 'Lyapunov Equations in The continuization of Linear systems, ' ACC conf . 1992 .
2.
Ahmad A. Mohammad, Doctoral Dissertation, ' M o d e l i n g Issues a n d the L y a p u n o v E q u a t i o n s in D y n a m i c C o n t r o l Systems,' The U n i v e r s i t y of Akron, A k r o n ,Ohio, 1992.
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
297
3.
H a m m a r l i n g , S.J., " N u m e r i c a l s o l u t i o n of the stable, n o n - n e g a t i v e d e f i n i t e L y a p u n o v e q u a t i o n , " IMA J. Numer. Anal., Vol. 2, pp. 303-323, 1982.
4.
Bitmead, R o b e r t R., " E x p l i c i t s o l u t i o n s of the d i s c r e t e - t i m e L y a p u n o v m a t r i x e q u a t i o n and K a l m a n Y a k o v i c h e q u a t i o n s , " IEEE Tran . Aut . .cont. , Vol AC-26, No. 6, pp. 1291-1294, Dec. 1981.
5.
Mori, Takehiro, Norio Fukuma, and Michioshi Kuwahara, "On the d i s c r e t e L y a p u n o v m a t r i x e q u a tion, " IEEE Trans . Aut . cont., . Vol. AC-27, No 2, pp. 463-464, A p r i l 1982.
6.
Wicks, M.A. and A s s i g n m e n t B a s e d on Trans. Aut. Control, A p r i l 1990.
7.
Wyman, B .F. and Sain, M.K, "On Dual Zero Spaces and I n v e r s e S y s t e m s , " IEEE Trans. Aut. Control, Vol. AC-31, NO. Ii, pp. 1053-1055, N o v e m b e r 1986.
8.
Yip, E.L. and Sincovec, R .F . , Controllability, and Observability Descriptor S y s t e m s , " IEEE T r a n s Control, Vol. AC--26, No. 3, pp. 1981.
9.
Young, N.J. , " F o r m u l a e for the solution of L y a p u n o v m a t r i x equations, " INT . J . Control, Vol 31, NO. I, p p . 1 5 9 - 1 7 9 , 1980.
I0 .
Zhou, Z., S h a y m a n M.A., and Tran, T.T., " S i n g u l a r Systems- A N e w A p p r o a c h in the Time Domain, " IEEE Trans. Aut. C o n t r o l , Vol. AC-32, NO. I, J a n u a r y 1987.
II.
Zietak, k., "On a P a r t i c u l a r Case of the i n c o n s i s tent L i n e a r M a t r i x E q u a t i o n A X + Y B = C , " L i n e a r Alg. and Appl., Vol. 66, pp. 249--258, April 1985.
12.
Zietak, k., " P r o p e r t i e s of the i p l - s o l u t i o n of the L i n e a r M a t r i x E q u a t i o n AX+YB=C, " IMA J of Num. Anal., Vol. 7, pp. 223--233, 1987.
13.
Zietak, k., "The C h e b y s h e v S o l u t i o n to the L i n e a r Matrix Equation AX+YB=C," Numerische Mathematik,
"G r a m m i an Decarlo, R.A., the L y a p u n o v E q u a t i o n , " IEEE Vol. 35, NO. 4, pp. 465-468,
"Solvability, of C o n t i n u o u s on A u t o m a t i c 702-707, June
298
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
Vol.
46, pp.
455--478,
1985.
14.
Zietak, k., "The lp--Solution of the Linear M a t r i x E q u a t i o n AX+YB=C, " Computing, Vol . 32, pp . 153162, 1984.
15.
Barnett, S. and Stability Theory,
16.
Barnett, Company,
17.
Gover, M.J.C. M a t r i x Theory,
18.
Kalman, R.E., " M a t h e m a t i c a l D e s c r i p t i o n of Linear I, D y n a m i c Systems, " SIAM J o u r n a l Control, Vol No. 3, pp. 152--192, 1963.
19.
Wonham, W.M., Linear Multivariable Control" a G e o m e t r i c Approach, 2nd Ed., Springer-Verlag, New York, 1979.
20.
Wonham, W.M., "On Pole A s s i g n m e n t in M u l t i - I n p u t C o n t r o l l a b l e S y s t e m s , " IEEE Trans. on A u t o m a t i c Control, Vol. AC-12, No. 6, pp. 660-665, D e c e m b e r 1967.
21.
Moore, B.C., "Principal Component Analysis in L i n e a r SystemsControllability, Observability, "IEEE Trans. Aut Control, and M o d e l reduction, Vol. AC-26, No. I, pp. 17-31, February 1981.
22.
Glover, Keith, "All optimal H a n k e l - n o r m approximations of linear m u l t i v a r i a b l e systems and their
Storey, C., M a t r i x M e t h o d s Barnes and Noble, Inc., 1970.
S., M a t r i c e s 1971.
in
VNR
Theory,
a n d S. Barnett, Applications C L A R E N D O N Press. Oxford, 1989.
L~ -error 6, pp.
Control
bounds, " INT . J . Control, 1115-1193, 1984.
Vol
39,
23.
Brogan, W.L., P r e n t i c e Hall, 07632.
24.
Kailath, T., Linear Systems, Prentice-Hall, E n g l e w o o d Cliffs, New Jersey, 1980.
25.
Lewis, F.L., Optimal New York, 1986.
in
of
NO.
M O D E R N C O N T R O L THEORY, 2nd ed., Inc., E n g l e w o o d Cliffs, New Jersey
Control,
Inc.,
John Wiley and Sons,
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
299
26.
Kalman, R.E., Ho, Y.C., and N a r e n d a , K.S., " C o n t r o l l a b i l i t y of Linear D y n a m i c a l Systems," in Contrib. Differ. Equations, Vol. I, pp. 189--213, 1962.
27.
Kuo, B.C., A u t o m a t i c C o n t r o l Systems, P r e n t i c e Hall, I4n4c., E n g l e w o o d Cliffs, New Jersey, 1987.
28.
Chen, C.T., Linear S y s t e m Theory and Design, P u b l i s h i n g Co., New York, New York, 1984.
29.
Taussky, O. and Wielandt, "On M a t r i x Function Ax'+ x'A' " Arch. R a t i o n a l Mech. Anal., Vol. 9, No. 7, pp. 9 3 - 9 6 , 1961.
30.
Ostrowski, A.M. and and Schneider, H., "Theorems on the i n e r t i a of g e n e r a l m a t r i c e s , " J . M a t h . Anal. Applic., Vol. 4, pp. 72-84, 1962.
31.
G a n t m a c k e r , F.R, The T h e o r y of Matrices, New York, Chelsa P u b l i s h i n g Co., 1960.
32.
Power, H.M., "A Note on the M a t r i x Equation A'LAL=-K, " IEEE Trans. Aut. Contrl, pp. 411-412, Aug. 1969.
33.
Power, H.M., " C a n o n i c a l form for the m a t r i c e s of l i n e a r d i s c r e t e - t i m e systems, " Proc. IEEE, Vol 116, No. 7, pp. 1245-1252, July 1969.
34.
Lancaster, Peter, " E x p l i c i t s o l u t i o n s of linear m a t r i x equation, " SIAM REVIEW, Vol. 12, No. 4, pp. 544-566, October 1970.
35.
Peters, G. and W i l k i n s o n , J.H., generalized eigenvalue problem," Anal., Vol. 7, pp. 479-492, 1970.
36.
Bartels, R.H. and G.W. Stewart, "Solution of m a t r i x e q u a t i o n A X + X B = C , " Comm. ACM, Vol pp. 820-826, 1972.
37.
Wu, Keqian, " M a t r i x W A + A ' W and it's a p p l i c a tions," INT . J Control, . . Vol . 47, . NO 6, pp 18731882, 1988.
32.
Moler, C.B. and Stewart, G. W., "An a l g o r i t h m for the g e n e r a l i z e d e i g e n v a l u e p r o b l e m Ax=Bx, "Tech.
HRW
Vol.l,
"Ax=Bx and the SIAM J. Numer. the 15,
300
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
Rept. CNA-32," Center Austin, TX, 1971.
for
numerical
analysis,
39.
Lewis, F.L., "Fundamental, Reachability, and O b s e r v a b i l i t y m a t r i c e s for d e s c r i p t o r s y s t e m s , " IEEE Trans. on A u t o m a t i c C o n t r o l Vol. AC-30, No. 5, pp. 502-505, 1985.
40.
Bender, D.J., "Lyapunov-Like Equations and Reachability /Observability Grammians for D e s c r i p t o r systems, "IEEE Trans. on A u t o m a t i c Control, Vol A C - 3 2 , No. 4, pp. 343--348, April 1987.
41.
Kalman, R.E., and J.E. Bertram, "Control S y s t e m Analysis a n d D e s i g n via the S e c o n d M e t h o d of Basic Lyapunov- I C o n t i n u o u s - T i m e Case, " AS ME J Eng., ser D. 86, pp. 51--60, 1964.
42.
Aldaheri, R a b a h W., "Model order r e d u c t i o n via real S c h u r - f o r m d e c o m p o s i t i o n , " INT . J . Control, Vol. 53, NO. 3, pp. 709-716, 1991.
43.
De Abreu-Garcia, J.A., "Balancing Techniques using J o r d a n F o r m R e a l i z a t i o n s , " Ph .D . D i s s e r t a t i o n , Queen's U n i v e r s i t y at Kingston, Kingston, Ontario, Canada, September 1986.
44.
Fernando, K.V. and Nicholson H., "On the Structure of B a l a n c e d and O t h e r P r i n c i p a l R e a l i z a t i o n s , " IEEE Trans. Aut. Control, Vol. AC-28, No. 2, pp. 228-231, April 1983.
45.
Fernando, K.V. and Nicholson H., "Singular Perturbational Model Reduction of Balanced S y s t e m s , " IEEE Trans. Aut. control, Vol. AC-27, No. 2, pp. 466-468, April 1982.
46.
Frankowska, H., "On c o n t r o l l a b i l i t y and o b s e r v ability of implicit systems," Syst. Control Lett., Vol. 14, No. 3, pp. 219-225, March 1990.
47.
Grag, W.B, Richel, L., and Dept. of Mathematics, Univ. of Kentucky, "on Singular Values of Hankel Operators of Finite Rank," Linear Algebra and it's Applications, Vol. 121, pp. 70-53, 1989.
48.
Hsu, C. S., Desai, U. B. , a n d " R e d u c t i o n of L a r g e - S c a l e Systems
Darden, R.J., via G e n e r a l i z e d
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
301
Grammians, " in Proc. 22nd IEEE Conf. on Decision and Control, pp. 1409-41410, December 1983. 49.
Kamei, E., Namba, H., Osaki, K., and Ohba, M., "Application of Reduced Order Model to Automotive Engine Control System," Trans. of the ASME, Vol. 109, pp. 232--237, September 1987.
50.
Klema, V.C. and Laub, A.J., "The Singular Value DecompositionIt 's Computation and Some Applications, " IEEE Trans . Control, Vol . AC-25, NO. 2, pp. 164-176, April 1980.
51.
Liu, YI and Anderson, B.D.O., "Singular perturbation approximation of b a l a n c e d systems," INT. J. Control, Vol. 50, NO.4, pp. 1379-1405, 1989.
52.
Pernebo, L. and Silverman, L.M., "Model Reduction via B a l a n c e d State Space R e p r e s e n t a t i o n s , " IEEE Trans. Aut. Control, Vol. AC-27, No. 2, April 1982.
53.
Laub, A.J., Heath, M.T., Paige, C.C., and Ward, R. C., "Computation of System Balancing Transformations and Other Applications of Simultaneous Diagonalization Algorithms," IEEE Trans. Aut. Control, Vol. AC-32, No. 2, February 1987.
54.
Francis, B., A C o u r s e in H ~ C o n t r o l Theory, Springer Verlag, New York, New York, 1987.
55.
Vidyasagar, M., "Normalized Coprime Factorizations for Nonstrictly Proper Systems, " IEEE Trans. Aut Control, Vol. 33, NO. 3, pp. 300-301, March 1988.
56.
Lambert, J.D., Computational Methods in Ordinary Differential Equations, Wiley, NY, 1973.
57.
Heij, C. , Deterministic Identification of Dynamical systems, Springer-Verlag, Berlin, Heidelberg, 1989.
58.
Goodwin, G.C. and Sin, K.S., Adaptive P r e d i c t i o n and Control, Prentice-Hall, Cliffs, New Jersey, 1984.
Filtering Englewood
302
A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A
59.
Santiago, J.M. and Jamshidi, M. , "On the Extensions of the B a l a n c e d A p p r o a c h for Model Reduction," Control-Theory an Advanced Technology, Vol. 2, NO.2, pp. 207-226, June 1986.
60.
Fernando, K.V. and S.J. Hammarling, "Computation of Balanced Realization from Transfer Functions," IEE Proc., Vol. 131, Pt.D, No. 5, pp. 203-206, September 1984.
61.
Kabamba,
"Balanced Gains and their L 2 Significance for Model Reduction," IEEE Trans. Aut. Control, Vol. AC-30, NO. 7, July 1985.
62.
Middleton, R.H. and G.C. Goodwin, Digital Control and Estimation- a Unified Approach, Prentice Hall, Englewood Cliffs, NJ 07632, 1990.
P.T. ,
AP P END IX CONT INUI ZAT ION
Given
1
EXAMPLE
Discrete
Time
System
(all matrices
are given
in
MATLAB notation) ad=[0.9000
0.I000;-0.i000
0.9000];bd=[l
I] ',cd=[l
i]; Balanced Discrete
time System
adb =[0.8500
0.1118;-0.1118
bdb =[1.4554
0.3438] ';cdb =[1.4554
Continuized
System via Bilinear
at =[-15.9543 bt =[1.5468
0.9500]; -0.3438];
Transform
12.3560;-12.3560
-4. 4153];
0.4413] ';ct =[154.6776
-44.1340];
dt =[-1.0497] ; Continuized Approach
Balanced
System
via
Lyapunov
Equations
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
303
segm =[7.7306 2.20581; acd =[-13.7002 9.0576;-9.0576 -2.67981; bcd =[14.5541 3.4383l;ccd =[14.5541
-3.43831;
Modified System Scaling Factor al=yd(200) /ycd(200)=0.9050 segmm=[6.9962 1.99621; acdm=[-15.1383 10.0084;-10.0084
-2.96111;
bcdm =[14.5541 3.43831';ccd =[14.5541
-3.43831;
304
AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA
S~p R e s p o n s e
15,
? -5 0 Fig.
50 100 Time=kT, T=.01
I. Step response, C T S v i a
~14
.
.
:
~ t..A.':, :-i...............
150 sec
200
Bilinear Transform
:
:
!. ..............
!..............
"~" 1D
,
[1
U
50
IDD
15D
Time=kT, T=. 01 sec
Fig. 2. Step response,
CTS via LBCT.
20D
CONTINUOUS
AND DISCRETE
915
TIME LYAPUNOV EQUATIONS
305
S~ep Response
~~~ f,o '
Fig.
0
50 I00 150 T J m e = k T , T = . 01 sec
3.
Step
2 .........
response, .
y
~
"~
9
: 9
, m"
I
... : ......... :.., . . . . .
% ~
,--2 ~
via Modified
LBCT
Error in Step Response
I
~0
CTS
200
.~ ("",
'~
/ o
;~
9
|
.
o o
~ o
:
:
.
~:
.
.. : _ -
:
-_ ~.~-: .--- . ~ . : - ,
:
.~-: ---
...!. .............. i. .............
9
~
:
:
.
:
9
,
.
1oo
15o
!
i
o Fig.
4.
5o
TJme=kT,
Step
response
i
T=.
errors
i
01
sec
2oo
306
A H M A D A. M O H A M M A D A N D J. A. D E A B R E U - G A R C I A
:=: 102
Frequency 9
0
,
,
9
9
9
Response, , , ,
9
9
9
9
Magnitude 9
9
, , ,
,
VS
,
9
9
,
,
W
,,
N
I
.
.
6,
o
\ I !
-rt r-t "r't " ' ~ Oo | !
9 10o
100 Fig.
o pa
1O, Frequency
5.
Frequency
Frequency
0
9. . . . .
9
w
"o oo
102 in Radians
response,
Response.
103
Magnitude
Phase
,"':";'t'ev
VS
%r
-v-
i
-50 o
~.,
~
-100
~
I
-o o . .o %=
I
"O00OOOO0 O
N -1so
I
.rl "r-I
:
-200 10o
.
.
.
.
.
.
.
,,
101
,
,
,
. . . . _ _
102
Frequency w in Radians Fig. 6. Frequency response, Phase
9
9
9 = .
.
103
CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS
Frequency Response,
102
9
,
.
,
.
o
, .
,
,
307
~agnitude VS
.
.
.
,
,
,
,
. . . .
,
. .
.
,
.
,
,
,
.
.
.
I
i
i
I
i
0 Dq I
9
6~
I0,
-
,~
-
~i,
-
o~
-
~
10o
-
0 ~
" -
!
-
I
.
10-1
.
.
.
.
, , ,
lOo
. . ,
Frequency Fig.
7. Frequency Frequeny
0 L~
. .
~
in
radians
response,
Response,
211...'_- .......
I
i
102
1 o1
9
Magnitude Phase
,
,
,
,
.
I
I
I
I
I/
,
VS
W
, ,
,
.
.
.
.
II
I
!
|
|
i
.
.
.
\'\
9 -20
i--I =L
o I,.4 o X:::
'
I
-40
-
-60
-
-80 -
-I00
m
100
i
I
I
I
I
Ig
I
I
101
Frequency Fig.
8. Frequency
102
w
in
radians
response,
Phase
!
i
1_
103
i
This Page Intentionally Left Blank
INDEX
Adaptive control systems, robust, discrete-time, s e e Robust adaptive control systems, discrete-time
B
Bartels and Stewart algorithm, Lyapunov equations, 267-269 Burstings robust adaptive control systems, 60-61 robust adaptive control systems and, 85-91 robust ultimate boundedness theorem, 64-66
C Circle criterion, discrete nonlinear feedback systems, 229-235 corollary 1,232-234 example, 234-235 lemma 1,230 theorem 1,231 Contraction mapping, discrete nonlinear feedback systems, 235-241 general considerations, 225-226 problem statement, 226-229 simultaneous Lyapunov sector, 236-241 evaluation, 237-241 example, 239-241 stability condition, 235-237 lemma 2, 236 theorem 3,236-237
Discrete-time systems extended output Loo constraint problem and, s e e Output Looconstraints, two--Riccati algorithm and nonlinear, feedback, 225-252 algebraic Riccati inequality approach, 242-251 absolute stability of multivariable systems, 242-245 lemma 3,243 theorem 4, 243-244 exponential stability, 248-251 lemma 4, 248-249 theorem 7, 249-250 theorem 8, 250 quadratic stability, 245-248 corollary 2, 247-248 theorem 6, 246-247 circle criterion, 229-235 corollary 1,232-234 example, 234-235 lemma 1,230 theorem 1, 231 contraction mapping approach, 235-241 simultaneous Lyapunov sector, 236-241 evaluation, 237-241 example, 239-241 stability condition, 235-237 lemma 2, 236 theorem 3, 236-237 general considerations, 225-226 problem statement, 226-229 periodically compensated, zero-placement
309
310
INDEX
approach, s e e Periodic controllers, discrete-time plants, zero-placement approach robust adapative control, s e e Robust adaptive control systems, discrete-time Duality theorem, periodic controllers, optimal and robust, mathematical preliminaries, 9-10
Fault detection and isolation systems, 171-224 general considerations, 171-174 model-based methods, 174-180 system dynamics, 174-175 fault detectability, 177 fault isolation, 177-180 fixed direction residual vector, 179-180 residual generators, 175-177 structured residuals set, 178-179 residual generation methods, 180-189 factorization methods, 187-188 observer-based approaches, 181-184 parameter estimation approach, 188 parity relation approaches, 184-186 relationship between generalized observer and parity relations, 186-187 stochastic systems, 188-189 robustness and, 189-216 conclusions, 215-217 decision-making, 212-215 adaptive thresholds, 212-215 fuzzy logic, 215 disturbance de-coupling principle, 190-201 eigenstructure assignment approach, 194-197 lemma 1,196 lemma 2, 196 theorem 2, 196-197 robust fault isolation schemes, 197-201 robust residual design via unknown input observers, 193-194 frequency domain approaches, 207-212 Marquez and Diduch robust observer, 207-209 multi-objective methods, 209-212 modelling errors, 191, 201-207 bilinear systems, 202-203 bounded uncertainty, 206-207 low rank approximation of distribution matrix, 205-206
model reduction, 203-204 noise and additive non-linearity, 202 optimal distribution matrix for varied operating points, 207 parameter perturbations, 204-205 Floquet theory, periodic controllers, di scretetime plants, zero-placement approach, 143-145 lemma 1, 143 - 144 theorem 2, 144-145 theorem 3, 145
Hammarling algorithm, Lyapunov equations, 269 Horowitz approach, periodic controllers, discrete-time plants, 169-170
K
Kronecker products, Lyapunov equations, 266-267
Lifting technique, periodic controllers discrete-time plants, zero-placement approach, 137-143 closed loop characteristic equation of twoperiodic systems, 142-143 linear, discrete, time-invariant (LDTI) transfer matrix, 137, 139-145 corollary 1.1, 141-142 theorem 1,139-141 optimal and robust, 11-18 disturbance rejection problem, 15-18 linear time invariant equivalence and, 11-15 multirate systems and, shift invariance equivalent, 44-52, 51-52 controller parametrization, 48--49 definitions, 44-46 example, 47-48 problem definition, 49-51 problem solution, 51-52 summary, 18-19 Linear, discrete, time-invariant (LDTI) transfer matrix, lifting technique, periodic controllers, 137, 139-145
INDEX Lyapunov equations continuous time and discrete-time, 253-307 applications, 272-296 H-infinity and robust control, 279-280 Lyapunov controllers, 273-274 model order reduction, 275-279 balancing technique, 275-279 properties, 278-279 new directions, 281-295 continuization problem, 282-295 algorithm, 284-288 appendix, 302-307 comparison with other techniques, 289-292 further developments, 292-293 numerical example, 293-295 preliminary, 282-284 theoretical aspects, 288-289 general considerations, 281-282 summary, 295-296 stability analysis, 274-275 system transformations, 280-281 continuization and discretization, 281 similarity transportations, 280-281 conclusions, 296 historical background, 254-264 controllability, observability, and minimality, 257-259 discrete Lyapunov equations, 263-265 grammians, 256-257 interpretation and importance of controllability and observability, 259-263 Lyapunov functions, 255-256 solutions, 264-272 descriptor Lyapunov-like equations, 271-272 discrete Lyapunov equations, 269-270 general considerations, 264-265 highlights, 265-269 Bartels and Stewart algorithm, 267-269 Hammarling algorithm, 269 Kronecker products, 266-267 contraction mapping approach, 236-241 evaluation, 237-241 example, 239-241 stability condition, 235-237 lemma 2, 236 theorem 3,236-237
311 M
Marques and Diduch robust observer, fault detection and isolation systems, 207-209 Multirate systems, optimal and robust controllers, 43-54; s e e a l s o Periodic controllers, optimal and robust lifting and shift invariant equivalence, 44-52 controller parametrization, 48-49 definitions, 44-46 example, 47-48 problem definition, 49-51 problem solution, 51-52 stability robustness, 53-54
Nehari's theorem, periodic controllers, optimal and robust, mathematical preliminaries, 11 Nevanlinna-Pick matrix method, periodic controllers, 168-169 Nonlinear systems, feedback, discrete-time, s e e Discrete-time systems, nonlinear, feedback
Output Loo constraints, two-Riccati algorithm and, 97-131 discrete-time version, 112-121 dynamic feeback case, 116-118 full order dynamic controller, 117 theorem 4.4, 117-118 example, 121-129 full order dynamic controller design, 125-128 state feedback controller design, 122-125 system matrices, 129-130 extended output Looconstraint problem, 114-115 lemma 4.1, 113-114 numerical algorithm, 118-121 full order dynamic controller, 119-121 lemma 4.5, 121 static measurement feedback, 119 static state feedback, 118-119 static measurement feedback case, 115-116 extended output Loo constraint problem solutions, 103-112
312
INDEX
dynamic feeback case, 106-109 full order dynamic controller, 107-108 theorem 3.4, 108-109 numerical algorithm, 109-112 full order dynamic controller, lll-ll2 lemma 3.8, 112 static measurement feedback, ll0-111 static state feedback, 1l0 static measurement feedback case, 103-106 lemma 3.3, 105-106 theorem 3.1, 104-105 statement, 100-103 lemma 2.1,100-101 lemma 2.3, 10 l - 102 general considerations, 97-100, 129 problem statement, 99-100
P
Periodic controllers discrete-time plants, zero-placement approach, 133-170 characteristic equation approach, 150-152 corollary 5.1, 152 completeness of controller form assumed, 167-168 conclusions and discussions, 161-162 controller configuration, 135-137 conventional approach, 149-150 Floquet theory, 143-145 lemma 1, 143-144 theorem 2, 144-145 theorem 3, 145 gain margin compensation, 153-161 bicausal plants, 153-154 strictly proper plants, 154-161 optimum gain margin compensation, 155-161 examples, 158-161 guidelines, 158 lemma 2, 156 theorem 6, 156-158 gain margin compensation using linear, discrete, time-invariant controller, 168-170 Horowitz approach, 169-170 Nevanlinna-Pick matrix method,
168-169 general considerations, 133-135 lifting technique, 137-143 closed loop characteristic equation of two-periodic systems, 142-143 linear, discrete, time-invariant clLDTI) transfer matrix, 137, 139-145 corollary 1.1,141-142 theorem 1,139-141 salient features of characteristic equation, 146-149 theorem 4, 148-149 single-input-single-output (SISO) periodic realization of transfer matrices, 163-169 lemma 3, 164-165 optimal and robust, 1-57 conclusions, 54-55 introduction, 1-2 12 tO 12disturbance rejection and LQG problems, 32-43 general considerations, 32-33 12to 2 robustness, 40-42 problem solution, 33-40 H2 problem, 39-40 H~176 problem, 34-39 summary, 42-43 lifting, 11-18 disturbance rejection problem, ].5-18 linear time invariant equivalence and, 11-15 summary, 18-19 mathematical preliminaries, 2-11 basic spaces, 3-5 fundamental connections, 8-9 genetic notation, 2-3 important theorems, 9-11 duality, 9-10 Nehari' s theorem, 11 projection theorem, 10 input-output characterization of linear systems, 5 - 8 causality, 5 - 6 stability, 6-7 exponential, 7 small gain theorem, 6 time (shift) invariance, 7-8 notational convention, 4 multirate systems and, 43-54 lifting and shift invariant equivalence, 44-52 controller parametrization, 48-49
INDEX definitions, 44-46 example, 47-48 problem definition, 49-51 problem solution, 51-52 stability robustness, 53-54 optimal 1~176 to 1~176 disturbance rejection, 19-32 general considerations, 19 problem solution, 19-29 examples, 28-29 robust stabilization, 30-32 summary, 32 Projection theorem, periodic controllers, optimal and robust, mathematical preliminaries, 10
R
Riccati equations algebraic Riccati inequality approach and, s e e Discrete-time systems, nonlinear, feedback output Looconstraints and, s e e Output Loo constraints, two-Riccati algorithm and Robust adaptive control systems; s e e a l s o Periodic controllers, discrete-time plants, zero-placement approach discrete-time, 59-96 algorithm, 69-85 lemma 4.1, 75-76 proof, 91-92
313
lemma 4.2, 76-77 proof, 92-95 self-stabilization, 79 theorem 4.1, 77-79 with burstings, 60-61 general considerations, 60-61 notation and terminology, 61-62 robust ultimate boundedness theorem, 62-69 bursting function, 64-66 Lyapunov function, 61, 63-69 theorem 3.1, 64-69 without burstings, 85-91 Robust ultimate bounded theorem, 62-69 Robustness, fault detection and isolation systems, s e e Fault detection and isolation systems, robustness and
Stochastic systems, fault detection and isolation systems, 188-189
Zero-placement, of periodically compensated discrete-time plants, s e e Periodic controllers, discrete-time plants, zero-placement approach
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